  
  [1X66 [33X[0;0YPolynomials and Rational Functions[133X[101X
  
  [33X[0;0YLet  [22XR[122X  be a commutative ring-with-one. We call a free associative algebra [22XA[122X
  over  [22XR[122X  a  [13Xpolynomial  ring[113X  over  [22XR[122X.  The  free generators of [22XA[122X are called
  [13Xindeterminates[113X (to avoid naming conflicts with the word [13Xvariables[113X which will
  be  used  to  denote  [5XGAP[105X variables only) , they are usually denoted by [22Xx_1,
  x_2, ...[122X. The number of indeterminates is called the [13Xrank[113X of [22XA[122X. The elements
  of   [22XA[122X  are  called  [13Xpolynomials[113X.  Products  of  indeterminates  are  called
  [13Xmonomials[113X,  every polynomial can be expressed as a finite sum of products of
  monomials  with ring elements in a form like [22Xr_{1,0} x_1 + r_{1,1} x_1 x_2 +
  r_{0,1} x_2 + ⋯[122X with [22Xr_{i,j} ∈ R[122X.[133X
  
  [33X[0;0YA  polynomial  ring  of  rank 1 is called an [13Xunivariate[113X polynomial ring, its
  elements are [13Xunivariate polynomials[113X.[133X
  
  [33X[0;0YPolynomial rings of smaller rank naturally embed in rings of higher rank; if
  [22XS[122X  is  a  subring  of  [22XR[122X then a polynomial ring over [22XS[122X naturally embeds in a
  polynomial  ring  over  [22XR[122X  of  the same rank. Note however that [5XGAP[105X does not
  consider  [22XR[122X as a subset of a polynomial ring over [22XR[122X; for example the zero of
  [22XR[122X ([22X0[122X) and the zero of the polynomial ring ([22X0x^0[122X) are different objects.[133X
  
  [33X[0;0YInternally,  indeterminates  are represented by positive integers, but it is
  possible  to give names to them to have them printed in a nicer way. Beware,
  however  that  there  is  not  necessarily  any  relation between the way an
  indeterminate  is  called  and  the  way it is printed. See section [14X66.1[114X for
  details.[133X
  
  [33X[0;0YIf  [22XR[122X  is  an  integral  domain, the polynomial ring [22XA[122X over [22XR[122X is an integral
  domain  as  well and one can therefore form its quotient field [22XQ[122X. This field
  is called a [13Xfield of rational functions[113X. Again [22XA[122X embeds naturally into [22XQ[122X and
  [5XGAP[105X  will perform this embedding implicitly. (In fact it implements the ring
  of  rational functions over [22XR[122X.) To avoid problems with leading coefficients,
  however, [22XR[122X must be a unique factorization domain.[133X
  
  
  [1X66.1 [33X[0;0YIndeterminates[133X[101X
  
  [33X[0;0YInternally,  indeterminates are created for a [13Xfamily[113X of objects (for example
  all elements of finite fields in characteristic [22X3[122X are in one family). Thus a
  variable  [21Xx[121X  over  the  rationals is also an [21Xx[121X over the integers, while an [21Xx[121X
  over [10XGF(3)[110X is different.[133X
  
  [33X[0;0YWithin  one  family,  every  indeterminate has a number [3Xnr[103X and as long as no
  other  names  have  been  assigned,  this indeterminate will be displayed as
  [21X[10Xx_[3Xnr[103X[10X[110X[121X. Indeterminate numbers can be arbitrary nonnegative integers.[133X
  
  [33X[0;0YIt  is  possible  to assign names to indeterminates; these names are strings
  and  only  provide  a  means  for printing the indeterminates in a nice way.
  Indeterminates that have not been assigned a name will be printed as [21X[10Xx_[3Xnr[103X[10X[110X[121X.[133X
  
  [33X[0;0Y(Because of this printing convention, the name [10Xx_[3Xnr[103X[10X[110X is interpreted specially
  to always denote the variable with internal number [3Xnr[103X.)[133X
  
  [33X[0;0YThe  indeterminate  names  have  not  necessarily  any relations to variable
  names:  this  means  that  an  indeterminate whose name is, say, [21X[10Xx[110X[121X cannot be
  accessed   using   the   variable  [10Xx[110X,  unless  [10Xx[110X  was  defined  to  be  that
  indeterminate.[133X
  
  [33X[0;0YWhen asking for indeterminates with certain names, [5XGAP[105X usually will take the
  first  (with  respect to the internal numbering) indeterminates that are not
  yet  named,  name  these  accordingly  and return them. Thus when asking for
  named  indeterminates,  no  relation between names and indeterminate numbers
  can            be            guaranteed.            The            attribute
  [10XIndeterminateNumberOfLaurentPolynomial([3Xindet[103X[10X)[110X  will return the number of the
  indeterminate [3Xindet[103X.[133X
  
  [33X[0;0YWhen  asked  to  create an indeterminate with a name that exists already for
  the  family,  [5XGAP[105X will by default return this existing indeterminate. If you
  explicitly  want a [13Xnew[113X indeterminate, distinct from the already existing one
  with  the  [13Xsame[113X name, you can add the [10Xnew[110X option to the function call. (This
  is in most cases not a good idea.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:=PolynomialRing(GF(3),["x","y","z"]);[127X[104X
    [4X[28XGF(3)[x,y,z][128X[104X
    [4X[25Xgap>[125X [27XList(IndeterminatesOfPolynomialRing(R),[127X[104X
    [4X[25X>[125X [27X  IndeterminateNumberOfLaurentPolynomial);[127X[104X
    [4X[28X[ 1, 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27XR:=PolynomialRing(GF(3),["z"]);[127X[104X
    [4X[28XGF(3)[z][128X[104X
    [4X[25Xgap>[125X [27XList(IndeterminatesOfPolynomialRing(R),[127X[104X
    [4X[25X>[125X [27X  IndeterminateNumberOfLaurentPolynomial);[127X[104X
    [4X[28X[ 3 ][128X[104X
    [4X[25Xgap>[125X [27XR:=PolynomialRing(GF(3),["x","y","z"]:new);[127X[104X
    [4X[28XGF(3)[x,y,z][128X[104X
    [4X[25Xgap>[125X [27XList(IndeterminatesOfPolynomialRing(R),[127X[104X
    [4X[25X>[125X [27X  IndeterminateNumberOfLaurentPolynomial);[127X[104X
    [4X[28X[ 4, 5, 6 ][128X[104X
    [4X[25Xgap>[125X [27XR:=PolynomialRing(GF(3),["z"]);[127X[104X
    [4X[28XGF(3)[z][128X[104X
    [4X[25Xgap>[125X [27XList(IndeterminatesOfPolynomialRing(R),[127X[104X
    [4X[25X>[125X [27X  IndeterminateNumberOfLaurentPolynomial);[127X[104X
    [4X[28X[ 3 ][128X[104X
  [4X[32X[104X
  
  
  [1X66.1-1 [33X[0;0YIndeterminate[133X[101X
  
  [29X[2XIndeterminate[102X( [3XR[103X[, [3Xnr[103X] ) [32X operation
  [29X[2XIndeterminate[102X( [3XR[103X[, [3Xname[103X][, [3Xavoid[103X] ) [32X operation
  [29X[2XIndeterminate[102X( [3Xfam[103X, [3Xnr[103X ) [32X operation
  [29X[2XX[102X( [3XR[103X[, [3Xnr[103X] ) [32X operation
  [29X[2XX[102X( [3XR[103X[, [3Xname[103X][, [3Xavoid[103X] ) [32X operation
  [29X[2XX[102X( [3Xfam[103X, [3Xnr[103X ) [32X operation
  
  [33X[0;0Yreturns  the  indeterminate number [3Xnr[103X over the ring [3XR[103X. If [3Xnr[103X is not given it
  defaults to 1. If the number is not specified a list [3Xavoid[103X of indeterminates
  may  be  given. The function will return an indeterminate that is guaranteed
  to  be  different  from  all the indeterminates in the list [3Xavoid[103X. The third
  usage returns an indeterminate called [3Xname[103X (also avoiding the indeterminates
  in [3Xavoid[103X if given).[133X
  
  [33X[0;0Y[2XX[102X is simply a synonym for [2XIndeterminate[102X. However, we do not recommend to use
  this synonym which is supported only for the backwards compatibility.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xx:=Indeterminate(GF(3),"x");[127X[104X
    [4X[28Xx[128X[104X
    [4X[25Xgap>[125X [27Xy:=X(GF(3),"y");z:=X(GF(3),"X");[127X[104X
    [4X[28Xy[128X[104X
    [4X[28XX[128X[104X
    [4X[25Xgap>[125X [27XX(GF(3),2);[127X[104X
    [4X[28Xy[128X[104X
    [4X[25Xgap>[125X [27XX(GF(3),"x_3");[127X[104X
    [4X[28XX[128X[104X
    [4X[25Xgap>[125X [27XX(GF(3),[y,z]);[127X[104X
    [4X[28Xx[128X[104X
  [4X[32X[104X
  
  [1X66.1-2 IndeterminateNumberOfUnivariateRationalFunction[101X
  
  [29X[2XIndeterminateNumberOfUnivariateRationalFunction[102X( [3Xrfun[103X ) [32X attribute
  
  [33X[0;0Yreturns  the  number  of  the indeterminate in which the univariate rational
  function  [3Xrfun[103X  is expressed. (This also provides a way to obtain the number
  of a given indeterminate.)[133X
  
  [33X[0;0YA  constant  rational function might not possess an indeterminate number. In
  this  case [2XIndeterminateNumberOfUnivariateRationalFunction[102X will default to a
  value  of 1. Therefore two univariate polynomials may be considered to be in
  the  same  univariate  polynomial ring if their indeterminates have the same
  number  or  one  if  of  them  is  constant.  (see  also [2XCIUnivPols[102X ([14X66.1-5[114X)
  and [2XIsLaurentPolynomialDefaultRep[102X ([14X66.21-7[114X)).[133X
  
  [1X66.1-3 IndeterminateOfUnivariateRationalFunction[101X
  
  [29X[2XIndeterminateOfUnivariateRationalFunction[102X( [3Xrfun[103X ) [32X attribute
  
  [33X[0;0Yreturns  the indeterminate in which the univariate rational function [3Xrfun[103X is
  expressed. (cf. [2XIndeterminateNumberOfUnivariateRationalFunction[102X ([14X66.1-2[114X).)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIndeterminateNumberOfUnivariateRationalFunction(z);[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XIndeterminateOfUnivariateRationalFunction(z^5+z);[127X[104X
    [4X[28XX[128X[104X
  [4X[32X[104X
  
  [1X66.1-4 IndeterminateName[101X
  
  [29X[2XIndeterminateName[102X( [3Xfam[103X, [3Xnr[103X ) [32X operation
  [29X[2XHasIndeterminateName[102X( [3Xfam[103X, [3Xnr[103X ) [32X operation
  [29X[2XSetIndeterminateName[102X( [3Xfam[103X, [3Xnr[103X, [3Xname[103X ) [32X operation
  
  [33X[0;0Y[2XSetIndeterminateName[102X  assigns  the  name  [3Xname[103X  to  indeterminate  [3Xnr[103X in the
  rational  functions  family [3Xfam[103X. It issues an error if the indeterminate was
  already named.[133X
  
  [33X[0;0Y[2XIndeterminateName[102X  returns  the name of the [3Xnr[103X-th indeterminate (and returns
  [9Xfail[109X if no name has been assigned).[133X
  
  [33X[0;0Y[2XHasIndeterminateName[102X   tests  whether  indeterminate  [3Xnr[103X  has  already  been
  assigned a name.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIndeterminateName(FamilyObj(x),2);[127X[104X
    [4X[28X"y"[128X[104X
    [4X[25Xgap>[125X [27XHasIndeterminateName(FamilyObj(x),4);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XSetIndeterminateName(FamilyObj(x),10,"bla");[127X[104X
    [4X[25Xgap>[125X [27XIndeterminate(GF(3),10);[127X[104X
    [4X[28Xbla[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAs  a  convenience there is a special method installed for [10XSetName[110X that will
  assign a name to an indeterminate.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:=Indeterminate(GF(3),5);[127X[104X
    [4X[28Xx_5[128X[104X
    [4X[25Xgap>[125X [27XSetName(a,"ah");[127X[104X
    [4X[25Xgap>[125X [27Xa^5+a;[127X[104X
    [4X[28Xah^5+ah[128X[104X
  [4X[32X[104X
  
  [1X66.1-5 CIUnivPols[101X
  
  [29X[2XCIUnivPols[102X( [3Xupol1[103X, [3Xupol2[103X ) [32X function
  
  [33X[0;0YThis  function  (whose  name  stands  for [21Xcommon indeterminate of univariate
  polynomials[121X)   takes  two  univariate  polynomials  as  arguments.  If  both
  polynomials  are given in the same indeterminate number [3Xindnum[103X (in this case
  they  are  [21Xcompatible[121X  as  univariate polynomials) it returns [3Xindnum[103X. In all
  other cases it returns [9Xfail[109X. [2XCIUnivPols[102X also accepts if either polynomial is
  constant  but formally expressed in another indeterminate, in this situation
  the indeterminate of the other polynomial is selected.[133X
  
  
  [1X66.2 [33X[0;0YOperations for Rational Functions[133X[101X
  
  [33X[0;0YThe rational functions form a field, therefore all arithmetic operations are
  applicable to rational functions.[133X
  
  [33X[0;0Y[10X[3Xf[103X[10X + [3Xg[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xf[103X[10X - [3Xg[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xf[103X[10X * [3Xg[103X[10X[110X[133X
  
  [33X[0;0Y[10X[3Xf[103X[10X / [3Xg[103X[10X[110X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xx:=Indeterminate(Rationals,1);;y:=Indeterminate(Rationals,2);;[127X[104X
    [4X[25Xgap>[125X [27Xf:=3+x*y+x^5;;g:=5+x^2*y+x*y^2;;[127X[104X
    [4X[25Xgap>[125X [27Xa:=g/f;[127X[104X
    [4X[28X(x_1^2*x_2+x_1*x_2^2+5)/(x_1^5+x_1*x_2+3)[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  the  quotient  [10X[3Xf[103X[10X/[3Xg[103X[10X[110X  of two polynomials might be represented as a
  rational  function  again.  If [3Xg[103X is known to divide [3Xf[103X the call [10XQuotient([3Xf[103X[10X,[3Xg[103X[10X)[110X
  (see [2XQuotient[102X ([14X56.1-9[114X)) should be used instead.[133X
  
  [33X[0;0Y[10X[3Xf[103X[10X mod [3Xg[103X[10X[110X[133X
  
  [33X[0;0YFor  two  Laurent  polynomials  [3Xf[103X  and [3Xg[103X, [10X[3Xf[103X[10X mod [3Xg[103X[10X[110X is the Euclidean remainder
  (see [2XEuclideanRemainder[102X ([14X56.6-4[114X)) of [3Xf[103X modulo [3Xg[103X.[133X
  
  [33X[0;0YAs  calculating  a  multivariate  Gcd can be expensive, it is not guaranteed
  that  rational functions will always be represented as a quotient of coprime
  polynomials.  In  certain unfortunate situations this might lead to a degree
  explosion.   To  ensure  cancellation  you  can  use  [2XGcd[102X  ([14X56.7-1[114X)  on  the
  [2XNumeratorOfRationalFunction[102X   ([14X66.4-2[114X)   and   [2XDenominatorOfRationalFunction[102X
  ([14X66.4-3[114X) values of a given rational function.[133X
  
  [33X[0;0YAll  polynomials  as  well  as  all  the  univariate polynomials in the same
  indeterminate  form  subrings  of  this field. If two rational functions are
  known  to be in the same subring, the result will be expressed as element in
  this subring.[133X
  
  
  [1X66.3 [33X[0;0YComparison of Rational Functions[133X[101X
  
  [33X[0;0Y[10X[3Xf[103X[10X = [3Xg[103X[10X[110X[133X
  
  [33X[0;0YTwo  rational  functions  [3Xf[103X  and  [3Xg[103X  are equal if the product [10XNumerator([3Xf[103X[10X) *
  Denominator([3Xg[103X[10X)[110X equals [10XNumerator([3Xg[103X[10X) * Denominator([3Xf[103X[10X)[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xx:=Indeterminate(Rationals,"x");;y:=Indeterminate(Rationals,"y");;[127X[104X
    [4X[25Xgap>[125X [27Xf:=3+x*y+x^5;;g:=5+x^2*y+x*y^2;;[127X[104X
    [4X[25Xgap>[125X [27Xa:=g/f;[127X[104X
    [4X[28X(x^2*y+x*y^2+5)/(x^5+x*y+3)[128X[104X
    [4X[25Xgap>[125X [27Xb:=(g*f)/(f^2);[127X[104X
    [4X[28X(x^7*y+x^6*y^2+5*x^5+x^3*y^2+x^2*y^3+3*x^2*y+3*x*y^2+5*x*y+15)/(x^10+2\[128X[104X
    [4X[28X*x^6*y+6*x^5+x^2*y^2+6*x*y+9)[128X[104X
    [4X[25Xgap>[125X [27Xa=b;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[10X[3Xf[103X[10X < [3Xg[103X[10X[110X[133X
  
  [33X[0;0YThe  ordering  of  rational functions is defined in several steps. Monomials
  (products   of   indeterminates)   are   sorted   first   by   degree,  then
  lexicographically   (with  [22Xx_1>x_2[122X)  (see [2XMonomialGrlexOrdering[102X  ([14X66.17-8[114X)).
  Products of monomials with ring elements ([21Xterms[121X) are compared first by their
  monomials and then by their coefficients.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xx>y;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xx^2*y<x*y^2;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xx*y<x^2*y;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xx^2*y < 5* y*x^2;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YPolynomials  are  compared by comparing the largest terms in turn until they
  differ.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xx+y<y;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xx<x+1;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YRational  functions  are compared by comparing the polynomial [10XNumerator([3Xf[103X[10X) *
  Denominator([3Xg[103X[10X)[110X  with  the  polynomial [10XNumerator([3Xg[103X[10X) * Denominator([3Xf[103X[10X)[110X. (As the
  ordering  of monomials used by [5XGAP[105X is invariant under multiplication this is
  independent of common factors in numerator and denominator.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf/g<g/f;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xf/g<(g*g)/(f*g);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor univariate polynomials this reduces to an ordering first by total degree
  and then lexicographically on the coefficients.[133X
  
  
  [1X66.4 [33X[0;0YProperties and Attributes of Rational Functions[133X[101X
  
  [33X[0;0YAll  these tests are applicable to [13Xevery[113X rational function. Depending on the
  internal  representation  of  the  rational  function, however some of these
  tests (in particular, univariateness) might be expensive in some cases.[133X
  
  [33X[0;0YFor  reasons  of performance within algorithms it can be useful to use other
  attributes,  which  give  a  slightly  more  technical  representation.  See
  section [14X66.20[114X for details.[133X
  
  [1X66.4-1 IsPolynomialFunction[101X
  
  [29X[2XIsPolynomialFunction[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsRationalFunction[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YA rational function is an element of the quotient field of a polynomial ring
  over  an  UFD.  It  is  represented  as  a  quotient of two polynomials, its
  numerator  (see [2XNumeratorOfRationalFunction[102X  ([14X66.4-2[114X))  and  its denominator
  (see [2XDenominatorOfRationalFunction[102X ([14X66.4-3[114X))[133X
  
  [33X[0;0YA polynomial function is an element of a polynomial ring (not necessarily an
  UFD), or a rational function.[133X
  
  [33X[0;0Y[5XGAP[105X considers [2XIsRationalFunction[102X as a subcategory of [2XIsPolynomialFunction[102X.[133X
  
  [1X66.4-2 NumeratorOfRationalFunction[101X
  
  [29X[2XNumeratorOfRationalFunction[102X( [3Xratfun[103X ) [32X attribute
  
  [33X[0;0Yreturns the numerator of the rational function [3Xratfun[103X.[133X
  
  [33X[0;0YAs  no  proper  multivariate  gcd  has  been implemented yet, numerators and
  denominators are not guaranteed to be reduced![133X
  
  [1X66.4-3 DenominatorOfRationalFunction[101X
  
  [29X[2XDenominatorOfRationalFunction[102X( [3Xratfun[103X ) [32X attribute
  
  [33X[0;0Yreturns the denominator of the rational function [3Xratfun[103X.[133X
  
  [33X[0;0YAs  no  proper  multivariate  gcd  has  been implemented yet, numerators and
  denominators are not guaranteed to be reduced![133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xx:=Indeterminate(Rationals,1);;y:=Indeterminate(Rationals,2);;[127X[104X
    [4X[25Xgap>[125X [27XDenominatorOfRationalFunction((x*y+x^2)/y);[127X[104X
    [4X[28Xy[128X[104X
    [4X[25Xgap>[125X [27XNumeratorOfRationalFunction((x*y+x^2)/y);[127X[104X
    [4X[28Xx^2+x*y[128X[104X
  [4X[32X[104X
  
  [1X66.4-4 IsPolynomial[101X
  
  [29X[2XIsPolynomial[102X( [3Xratfun[103X ) [32X property
  
  [33X[0;0YA  polynomial  is  a  rational  function  whose  denominator is one. (If the
  coefficients  family  forms  a  field  this is equivalent to the denominator
  being constant.)[133X
  
  [33X[0;0YIf  the  base  family  is not a field, it may be impossible to represent the
  quotient  of  a  polynomial  by a ring element as a polynomial again, but it
  will have to be represented as a rational function.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsPolynomial((x*y+x^2*y^3)/y);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsPolynomial((x*y+x^2)/y);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X66.4-5 AsPolynomial[101X
  
  [29X[2XAsPolynomial[102X( [3Xpoly[103X ) [32X attribute
  
  [33X[0;0YIf  [3Xpoly[103X  is a rational function that is a polynomial this attribute returns
  an  equal  rational function [22Xp[122X such that [22Xp[122X is equal to its numerator and the
  denominator of [22Xp[122X is one.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAsPolynomial((x*y+x^2*y^3)/y);[127X[104X
    [4X[28Xx^2*y^2+x[128X[104X
  [4X[32X[104X
  
  [1X66.4-6 IsUnivariateRationalFunction[101X
  
  [29X[2XIsUnivariateRationalFunction[102X( [3Xratfun[103X ) [32X property
  
  [33X[0;0YA  rational  function is univariate if its numerator and its denominator are
  both   polynomials   in   the   same   one   indeterminate.   The  attribute
  [2XIndeterminateNumberOfUnivariateRationalFunction[102X  ([14X66.1-2[114X)  can  be  used  to
  obtain the number of this common indeterminate.[133X
  
  [1X66.4-7 CoefficientsOfUnivariateRationalFunction[101X
  
  [29X[2XCoefficientsOfUnivariateRationalFunction[102X( [3Xrfun[103X ) [32X attribute
  
  [33X[0;0Yif  [3Xrfun[103X  is a univariate rational function, this attribute returns a list [10X[
  [3Xncof[103X[10X,  [3Xdcof[103X[10X,  [3Xval[103X[10X  ][110X where [3Xncof[103X and [3Xdcof[103X are coefficient lists of univariate
  polynomials [3Xn[103X and [3Xd[103X and a valuation [3Xval[103X such that [22X[3Xrfun[103X = x^[3Xval[103X ⋅ [3Xn[103X / [3Xd[103X[122X where
  [22Xx[122X      is      the     variable     with     the     number     given     by
  [2XIndeterminateNumberOfUnivariateRationalFunction[102X   ([14X66.1-2[114X).   Numerator  and
  denominator are guaranteed to be cancelled.[133X
  
  [1X66.4-8 IsUnivariatePolynomial[101X
  
  [29X[2XIsUnivariatePolynomial[102X( [3Xratfun[103X ) [32X property
  
  [33X[0;0YA univariate polynomial is a polynomial in only one indeterminate.[133X
  
  [1X66.4-9 CoefficientsOfUnivariatePolynomial[101X
  
  [29X[2XCoefficientsOfUnivariatePolynomial[102X( [3Xpol[103X ) [32X attribute
  
  [33X[0;0Y[2XCoefficientsOfUnivariatePolynomial[102X  returns  the  coefficient  list  of  the
  polynomial [3Xpol[103X, sorted in ascending order. (It returns the empty list if [3Xpol[103X
  is 0.)[133X
  
  [1X66.4-10 IsLaurentPolynomial[101X
  
  [29X[2XIsLaurentPolynomial[102X( [3Xratfun[103X ) [32X property
  
  [33X[0;0YA  Laurent polynomial is a univariate rational function whose denominator is
  a monomial. Therefore every univariate polynomial is a Laurent polynomial.[133X
  
  [33X[0;0YThe  attribute  [2XCoefficientsOfLaurentPolynomial[102X  ([14X66.13-2[114X)  gives  a compact
  representation as Laurent polynomial.[133X
  
  [1X66.4-11 IsConstantRationalFunction[101X
  
  [29X[2XIsConstantRationalFunction[102X( [3Xratfun[103X ) [32X property
  
  [33X[0;0YA  constant  rational function is a function whose numerator and denominator
  are polynomials of degree 0.[133X
  
  [1X66.4-12 IsPrimitivePolynomial[101X
  
  [29X[2XIsPrimitivePolynomial[102X( [3XF[103X, [3Xpol[103X ) [32X operation
  
  [33X[0;0YFor  a  univariate  polynomial  [3Xpol[103X of degree [22Xd[122X in the indeterminate [22XX[122X, with
  coefficients in a finite field [3XF[103X with [22Xq[122X elements, say, [2XIsPrimitivePolynomial[102X
  returns [9Xtrue[109X if[133X
  
  [31X1[131X   [33X[0;6Y[3Xpol[103X divides [22XX^{q^d-1} - 1[122X, and[133X
  
  [31X2[131X   [33X[0;6Yfor each prime divisor [22Xp[122X of [22Xq^d - 1[122X, [3Xpol[103X does not divide [22XX^{(q^d-1)/p}
        - 1[122X,[133X
  
  [33X[0;0Yand [9Xfalse[109X otherwise.[133X
  
  [1X66.4-13 SplittingField[101X
  
  [29X[2XSplittingField[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0Yreturns  the  smallest  field  which  contains the coefficients of [3Xf[103X and the
  roots of [3Xf[103X.[133X
  
  
  [1X66.5 [33X[0;0YUnivariate Polynomials[133X[101X
  
  [33X[0;0YSome  of the operations are actually defined on the larger domain of Laurent
  polynomials  (see  [14X66.13[114X).  For  this section you can simply ignore the word
  [21XLaurent[121X if it occurs in a description.[133X
  
  [1X66.5-1 UnivariatePolynomial[101X
  
  [29X[2XUnivariatePolynomial[102X( [3Xring[103X, [3Xcofs[103X[, [3Xind[103X] ) [32X operation
  
  [33X[0;0Yconstructs  an univariate polynomial over the ring [3Xring[103X in the indeterminate
  [3Xind[103X with the coefficients given by [3Xcoefs[103X.[133X
  
  [1X66.5-2 UnivariatePolynomialByCoefficients[101X
  
  [29X[2XUnivariatePolynomialByCoefficients[102X( [3Xfam[103X, [3Xcofs[103X, [3Xind[103X ) [32X operation
  
  [33X[0;0Yconstructs  an univariate polynomial over the coefficients family [3Xfam[103X and in
  the  indeterminate  [3Xind[103X  with the coefficients given by [3Xcoefs[103X. This function
  should  be  used  in  algorithms to create polynomials as it avoids overhead
  associated with [2XUnivariatePolynomial[102X ([14X66.5-1[114X).[133X
  
  [1X66.5-3 DegreeOfLaurentPolynomial[101X
  
  [29X[2XDegreeOfLaurentPolynomial[102X( [3Xpol[103X ) [32X attribute
  
  [33X[0;0YThe  degree of a univariate (Laurent) polynomial [3Xpol[103X is the largest exponent
  [22Xn[122X of a monomial [22Xx^n[122X of [3Xpol[103X. The degree of a zero polynomial is defined to be
  [10X-infinity[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xp:=UnivariatePolynomial(Rationals,[1,2,3,4],1);[127X[104X
    [4X[28X4*x^3+3*x^2+2*x+1[128X[104X
    [4X[25Xgap>[125X [27XUnivariatePolynomialByCoefficients(FamilyObj(1),[9,2,3,4],73);[127X[104X
    [4X[28X4*x_73^3+3*x_73^2+2*x_73+9[128X[104X
    [4X[25Xgap>[125X [27XCoefficientsOfUnivariatePolynomial(p);[127X[104X
    [4X[28X[ 1, 2, 3, 4 ][128X[104X
    [4X[25Xgap>[125X [27XDegreeOfLaurentPolynomial(p);[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XDegreeOfLaurentPolynomial(Zero(p));[127X[104X
    [4X[28X-infinity[128X[104X
    [4X[25Xgap>[125X [27XIndeterminateNumberOfLaurentPolynomial(p);[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XIndeterminateOfLaurentPolynomial(p);[127X[104X
    [4X[28Xx[128X[104X
  [4X[32X[104X
  
  [1X66.5-4 RootsOfPolynomial[101X
  
  [29X[2XRootsOfPolynomial[102X( [[3XR[103X, ][3Xp[103X ) [32X function
  
  [33X[0;0YFor a univariate polynomial [3Xp[103X, this function returns all roots of [3Xp[103X over the
  ring  [3XR[103X.  If the ring is not specified, it defaults to the ring specified by
  the coefficients of [3Xp[103X via [2XDefaultRing[102X ([14X56.1-3[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xx:=X(Rationals,"x");;p:=x^4-1;    [127X[104X
    [4X[28Xx^4-1[128X[104X
    [4X[25Xgap>[125X [27XRootsOfPolynomial(p);[127X[104X
    [4X[28X[ 1, -1 ][128X[104X
    [4X[25Xgap>[125X [27XRootsOfPolynomial(CF(4),p);[127X[104X
    [4X[28X[ 1, -1, E(4), -E(4) ][128X[104X
  [4X[32X[104X
  
  [1X66.5-5 RootsOfUPol[101X
  
  [29X[2XRootsOfUPol[102X( [[3Xfield[103X, ][3Xupol[103X ) [32X function
  
  [33X[0;0YThis  function returns a list of all roots of the univariate polynomial [3Xupol[103X
  in  its  default  domain. If the optional argument [3Xfield[103X is a field then the
  roots  in  this  field are computed. If [3Xfield[103X is the string [10X"split"[110X then the
  splitting field of the polynomial is taken.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRootsOfUPol(50-45*x-6*x^2+x^3);[127X[104X
    [4X[28X[ 10, 1, -5 ][128X[104X
  [4X[32X[104X
  
  [1X66.5-6 QuotRemLaurpols[101X
  
  [29X[2XQuotRemLaurpols[102X( [3Xleft[103X, [3Xright[103X, [3Xmode[103X ) [32X function
  
  [33X[0;0YThis  internal  function  for  euclidean  division  of polynomials takes two
  polynomials [3Xleft[103X and [3Xright[103X and computes their quotient. No test is performed
  whether  the  arguments  indeed  are  polynomials.  Depending on the integer
  variable  [3Xmode[103X,  which  may  take  values in a range from 1 to 4, it returns
  respectively:[133X
  
  [31X1[131X   [33X[0;6Ythe quotient (there might be some remainder),[133X
  
  [31X2[131X   [33X[0;6Ythe remainder,[133X
  
  [31X3[131X   [33X[0;6Ya list [10X[[3Xq[103X[10X,[3Xr[103X[10X][110X of quotient and remainder,[133X
  
  [31X4[131X   [33X[0;6Ythe quotient if there is no remainder and [9Xfail[109X otherwise.[133X
  
  [1X66.5-7 UnivariatenessTestRationalFunction[101X
  
  [29X[2XUnivariatenessTestRationalFunction[102X( [3Xf[103X ) [32X function
  
  [33X[0;0Ytakes  a  rational  function  [3Xf[103X  and tests whether it is univariate rational
  function  (or  even  a Laurent polynomial). It returns a list [10X[isunivariate,
  indet, islaurent, cofs][110X.[133X
  
  [33X[0;0YIf  [3Xf[103X  is a univariate rational function then [10Xisunivariate[110X is [9Xtrue[109X and [10Xindet[110X
  is the number of the appropriate indeterminate.[133X
  
  [33X[0;0YFurthermore,  if  [3Xf[103X is a Laurent polynomial, then [10Xislaurent[110X is also [9Xtrue[109X. In
  this   case   the  fourth  entry,  [10Xcofs[110X,  is  the  value  of  the  attribute
  [2XCoefficientsOfLaurentPolynomial[102X ([14X66.13-2[114X) for [3Xf[103X.[133X
  
  [33X[0;0YIf  [10Xisunivariate[110X  is  [9Xtrue[109X but [10Xislaurent[110X is [9Xfalse[109X, then [10Xcofs[110X is the value of
  the attribute [2XCoefficientsOfUnivariateRationalFunction[102X ([14X66.4-7[114X) for [3Xf[103X.[133X
  
  [33X[0;0YOtherwise,  each entry of the returned list is equal to [9Xfail[109X. As there is no
  proper  multivariate  gcd,  this  may  also happen for the rational function
  which may be reduced to univariate (see example).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XUnivariatenessTestRationalFunction( 50-45*x-6*x^2+x^3 );[127X[104X
    [4X[28X[ true, 1, true, [ [ 50, -45, -6, 1 ], 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XUnivariatenessTestRationalFunction( (-6*y^2+y^3) / (y+1) );[127X[104X
    [4X[28X[ true, 2, false, [ [ -6, 1 ], [ 1, 1 ], 2 ] ][128X[104X
    [4X[25Xgap>[125X [27XUnivariatenessTestRationalFunction( (-6*y^2+y^3) / (x+1));[127X[104X
    [4X[28X[ fail, fail, fail, fail ][128X[104X
    [4X[25Xgap>[125X [27XUnivariatenessTestRationalFunction( ((y+2)*(x+1)) / ((y-1)*(x+1)) );[127X[104X
    [4X[28X[ fail, fail, fail, fail ][128X[104X
  [4X[32X[104X
  
  [1X66.5-8 InfoPoly[101X
  
  [29X[2XInfoPoly[102X[32X info class
  
  [33X[0;0Yis the info class for univariate polynomials.[133X
  
  [33X[0;0YWe  remark  that  some functions for multivariate polynomials (which will be
  defined  in the following sections) permit a different syntax for univariate
  polynomials  which  drops  the  requirement  to  specify  the indeterminate.
  Examples  are  [2XValue[102X  ([14X66.7-1[114X),  [2XDiscriminant[102X ([14X66.6-6[114X), [2XDerivative[102X ([14X66.6-5[114X),
  [2XLeadingCoefficient[102X ([14X66.6-3[114X) and [2XLeadingMonomial[102X ([14X66.6-4[114X):[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xp:=UnivariatePolynomial(Rationals,[1,2,3,4],1);[127X[104X
    [4X[28X4*x^3+3*x^2+2*x+1[128X[104X
    [4X[25Xgap>[125X [27XValue(p,Z(5));[127X[104X
    [4X[28XZ(5)^2[128X[104X
    [4X[25Xgap>[125X [27XLeadingCoefficient(p);[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XDerivative(p);[127X[104X
    [4X[28X12*x^2+6*x+2[128X[104X
  [4X[32X[104X
  
  
  [1X66.6 [33X[0;0YPolynomials as Univariate Polynomials in one Indeterminate[133X[101X
  
  [1X66.6-1 DegreeIndeterminate[101X
  
  [29X[2XDegreeIndeterminate[102X( [3Xpol[103X, [3Xind[103X ) [32X operation
  
  [33X[0;0Yreturns   the  degree  of  the  polynomial  [3Xpol[103X  in  the  indeterminate  (or
  indeterminate number) [3Xind[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf:=x^5+3*x*y+9*y^7+4*y^5*x+3*y+2;[127X[104X
    [4X[28X9*y^7+4*x*y^5+x^5+3*x*y+3*y+2[128X[104X
    [4X[25Xgap>[125X [27XDegreeIndeterminate(f,1);[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XDegreeIndeterminate(f,y);[127X[104X
    [4X[28X7[128X[104X
  [4X[32X[104X
  
  [1X66.6-2 PolynomialCoefficientsOfPolynomial[101X
  
  [29X[2XPolynomialCoefficientsOfPolynomial[102X( [3Xpol[103X, [3Xind[103X ) [32X operation
  
  [33X[0;0Y[2XPolynomialCoefficientsOfPolynomial[102X   returns  the  coefficient  list  (whose
  entries  are polynomials not involving the indeterminate [3Xind[103X) describing the
  polynomial  [3Xpol[103X viewed as a polynomial in [3Xind[103X. Instead of the indeterminate,
  [3Xind[103X can also be an indeterminate number.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPolynomialCoefficientsOfPolynomial(f,2);[127X[104X
    [4X[28X[ x^5+2, 3*x+3, 0, 0, 0, 4*x, 0, 9 ][128X[104X
  [4X[32X[104X
  
  [1X66.6-3 LeadingCoefficient[101X
  
  [29X[2XLeadingCoefficient[102X( [3Xpol[103X ) [32X operation
  
  [33X[0;0Yreturns  the  leading  coefficient  (that  is the coefficient of the leading
  monomial, see [2XLeadingMonomial[102X ([14X66.6-4[114X)) of the polynomial [3Xpol[103X.[133X
  
  [1X66.6-4 LeadingMonomial[101X
  
  [29X[2XLeadingMonomial[102X( [3Xpol[103X ) [32X function
  
  [33X[0;0Yreturns  the  leading  monomial  (with  respect  to  the  ordering  given by
  [2XMonomialExtGrlexLess[102X  ([14X66.17-14[114X)) of the polynomial [3Xpol[103X as a list containing
  indeterminate numbers and exponents.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLeadingCoefficient(f,1);[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XLeadingCoefficient(f,2);[127X[104X
    [4X[28X9[128X[104X
    [4X[25Xgap>[125X [27XLeadingMonomial(f);[127X[104X
    [4X[28X[ 2, 7 ][128X[104X
    [4X[25Xgap>[125X [27XLeadingCoefficient(f);[127X[104X
    [4X[28X9[128X[104X
  [4X[32X[104X
  
  [1X66.6-5 Derivative[101X
  
  [29X[2XDerivative[102X( [3Xratfun[103X[, [3Xind[103X] ) [32X attribute
  
  [33X[0;0YIf  [3Xratfun[103X  is  a  univariate  rational function then [2XDerivative[102X returns the
  [13Xderivative[113X of [3Xufun[103X by its indeterminate. For a rational function [3Xratfun[103X, the
  derivative  by  the  indeterminate  [3Xind[103X  is  returned,  regarding  [3Xratfun[103X as
  univariate  in [3Xind[103X. Instead of the desired indeterminate, also the number of
  this indeterminate can be given as [3Xind[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDerivative(f,2);[127X[104X
    [4X[28X63*y^6+20*x*y^4+3*x+3[128X[104X
  [4X[32X[104X
  
  [1X66.6-6 Discriminant[101X
  
  [29X[2XDiscriminant[102X( [3Xpol[103X[, [3Xind[103X] ) [32X operation
  
  [33X[0;0YIf [3Xpol[103X is a univariate polynomial then [2XDiscriminant[102X returns the [13Xdiscriminant[113X
  of  [3Xpol[103X by its indeterminate. The two-argument form returns the discriminant
  of  a  polynomial  [3Xpol[103X  by  the  indeterminate  number [3Xind[103X, regarding [3Xpol[103X as
  univariate  in  this indeterminate. Instead of the indeterminate number, the
  indeterminate itself can also be given as [3Xind[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDiscriminant(f,1);[127X[104X
    [4X[28X20503125*y^28+262144*y^25+27337500*y^22+19208040*y^21+1474560*y^17+136\[128X[104X
    [4X[28X68750*y^16+18225000*y^15+6075000*y^14+1105920*y^13+3037500*y^10+648972\[128X[104X
    [4X[28X0*y^9+4050000*y^8+900000*y^7+62208*y^5+253125*y^4+675000*y^3+675000*y^\[128X[104X
    [4X[28X2+300000*y+50000[128X[104X
    [4X[25Xgap>[125X [27XDiscriminant(f,1) = Discriminant(f,x);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X66.6-7 Resultant[101X
  
  [29X[2XResultant[102X( [3Xpol1[103X, [3Xpol2[103X, [3Xind[103X ) [32X operation
  
  [33X[0;0Ycomputes  the resultant of the polynomials [3Xpol1[103X and [3Xpol2[103X with respect to the
  indeterminate [3Xind[103X, or indeterminate number [3Xind[103X. The resultant considers [3Xpol1[103X
  and  [3Xpol2[103X  as  univariate in [3Xind[103X and returns an element of the corresponding
  base ring (which might be a polynomial ring).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XResultant(x^4+y,y^4+x,1);[127X[104X
    [4X[28Xy^16+y[128X[104X
    [4X[25Xgap>[125X [27XResultant(x^4+y,y^4+x,2);[127X[104X
    [4X[28Xx^16+x[128X[104X
  [4X[32X[104X
  
  
  [1X66.7 [33X[0;0YMultivariate Polynomials[133X[101X
  
  
  [1X66.7-1 [33X[0;0YValue[133X[101X
  
  [29X[2XValue[102X( [3Xratfun[103X, [3Xindets[103X, [3Xvals[103X[, [3Xone[103X] ) [32X operation
  [29X[2XValue[102X( [3Xupol[103X, [3Xvalue[103X[, [3Xone[103X] ) [32X operation
  
  [33X[0;0YThe  first  variant  takes  a  rational  function [3Xratfun[103X and specializes the
  indeterminates  given  in  [3Xindets[103X to the values given in [3Xvals[103X, replacing the
  [22Xi[122X-th  entry  in  [3Xindets[103X  by  the  [22Xi[122X-th entry in [3Xvals[103X. If this specialization
  results  in  a  constant  polynomial,  an element of the coefficient ring is
  returned.  If  the specialization would specialize the denominator of [3Xratfun[103X
  to zero, an error is raised.[133X
  
  [33X[0;0YA  variation  is  the  evaluation at elements of another ring [22XR[122X, for which a
  multiplication  with elements of the coefficient ring of [3Xratfun[103X are defined.
  In  this  situation  the  identity  element  of  [22XR[122X may be given by a further
  argument [3Xone[103X which will be used for [22Xx^0[122X for any specialized indeterminate [22Xx[122X.[133X
  
  [33X[0;0YThe second version takes an univariate rational function and specializes the
  value  of  its  indeterminate to [3Xval[103X. Again, an optional argument [3Xone[103X may be
  given.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XValue(x*y+y+x^7,[x,y],[5,7]);[127X[104X
    [4X[28X78167[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that the default values for [3Xone[103X can lead to different results than one
  would  expect:  For  example  for  a  matrix [22XM[122X, the values [22XM+M^0[122X and [22XM+1[122X are
  [13Xdifferent[113X.  As  [2XValue[102X  defaults  to  the  one  of the coefficient ring, when
  evaluating matrices in polynomials always the correct [3Xone[103X should be given![133X
  
  
  [1X66.8 [33X[0;0YMinimal Polynomials[133X[101X
  
  [1X66.8-1 MinimalPolynomial[101X
  
  [29X[2XMinimalPolynomial[102X( [3XR[103X, [3Xelm[103X[, [3Xind[103X] ) [32X operation
  
  [33X[0;0Yreturns  the  [13Xminimal  polynomial[113X  of  [3Xelm[103X over the ring [3XR[103X, expressed in the
  indeterminate number [3Xind[103X. If [3Xind[103X is not given, it defaults to 1.[133X
  
  [33X[0;0YThe  minimal  polynomial  is  the  monic  polynomial of smallest degree with
  coefficients in [3XR[103X that has value zero at [3Xelm[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMinimalPolynomial(Rationals,[[2,0],[0,2]]);[127X[104X
    [4X[28Xx-2[128X[104X
  [4X[32X[104X
  
  
  [1X66.9 [33X[0;0YCyclotomic Polynomials[133X[101X
  
  [1X66.9-1 CyclotomicPolynomial[101X
  
  [29X[2XCyclotomicPolynomial[102X( [3XF[103X, [3Xn[103X ) [32X function
  
  [33X[0;0Yis the [3Xn[103X-th cyclotomic polynomial over the ring [3XF[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCyclotomicPolynomial(Rationals,5);[127X[104X
    [4X[28Xx^4+x^3+x^2+x+1[128X[104X
  [4X[32X[104X
  
  
  [1X66.10 [33X[0;0YPolynomial Factorization[133X[101X
  
  [33X[0;0YAt the moment [5XGAP[105X provides only methods to factorize polynomials over finite
  fields   (see   Chapter [14X59[114X),   over  subfields  of  cyclotomic  fields  (see
  Chapter [14X60[114X), and over algebraic extensions of these (see Chapter [14X67[114X).[133X
  
  [1X66.10-1 Factors[101X
  
  [29X[2XFactors[102X( [[3XR[103X, ][3Xpoly[103X[, [3Xopt[103X] ) [32X method
  
  [33X[0;0Yreturns  a  list  of  the  irreducible factors of the polynomial [3Xpoly[103X in the
  polynomial  ring  [3XR[103X.  (That  is  factors over the [2XCoefficientsRing[102X ([14X66.15-3[114X)
  value of [3XR[103X.)[133X
  
  [33X[0;0YFor  univariate  factorizations,  it  is  possible to pass a record [3Xopt[103X as a
  third argument. This record can contain the following components:[133X
  
  [8X[10Xonlydegs[110X[108X
        [33X[0;6Yis  a  set  of  positive  integers. The factorization assumes that all
        irreducible factors have a degree in this set.[133X
  
  [8X[10Xstopdegs[110X[108X
        [33X[0;6Yis  a  set  of  positive  integers. The factorization will stop once a
        factor  of  degree  in  [10Xstopdegs[110X  has  been  found and will return the
        factorization found so far.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf:= CyclotomicPolynomial( GF(2), 7 );[127X[104X
    [4X[28Xx_1^6+x_1^5+x_1^4+x_1^3+x_1^2+x_1+Z(2)^0[128X[104X
    [4X[25Xgap>[125X [27XFactors( f );[127X[104X
    [4X[28X[ x_1^3+x_1+Z(2)^0, x_1^3+x_1^2+Z(2)^0 ][128X[104X
    [4X[25Xgap>[125X [27XFactors( PolynomialRing( GF(8) ), f );[127X[104X
    [4X[28X[ x_1+Z(2^3), x_1+Z(2^3)^2, x_1+Z(2^3)^3, x_1+Z(2^3)^4, x_1+Z(2^3)^5, [128X[104X
    [4X[28X  x_1+Z(2^3)^6 ][128X[104X
    [4X[25Xgap>[125X [27Xf:= MinimalPolynomial( Rationals, E(4) );[127X[104X
    [4X[28Xx^2+1[128X[104X
    [4X[25Xgap>[125X [27XFactors( f );[127X[104X
    [4X[28X[ x^2+1 ][128X[104X
    [4X[25Xgap>[125X [27XFactors( PolynomialRing( Rationals ), f );[127X[104X
    [4X[28X[ x^2+1 ][128X[104X
    [4X[25Xgap>[125X [27XFactors( PolynomialRing( CF(4) ), f );[127X[104X
    [4X[28X[ x+(-E(4)), x+E(4) ][128X[104X
  [4X[32X[104X
  
  [1X66.10-2 FactorsSquarefree[101X
  
  [29X[2XFactorsSquarefree[102X( [3Xpring[103X, [3Xupol[103X, [3Xopt[103X ) [32X operation
  
  [33X[0;0Yreturns a factorization of the squarefree, monic, univariate polynomial [3Xupol[103X
  in  the  polynomial  ring  [3Xpring[103X;  [3Xopt[103X  must be a (possibly empty) record of
  options.  [3Xupol[103X  must  not  have zero as a root. This function is used by the
  factoring algorithms.[133X
  
  [33X[0;0YThe  current  method  for  multivariate  factorization reduces to univariate
  factorization  by use of a reduction homomorphism of the form [22Xf(x_1,x_2,x_3)
  ↦ f(x,x^p,x^{p^2})[122X. It can be very time intensive for larger degrees.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFactors(x^10-y^10);[127X[104X
    [4X[28X[ x-y, x+y, x^4-x^3*y+x^2*y^2-x*y^3+y^4, x^4+x^3*y+x^2*y^2+x*y^3+y^4 ][128X[104X
  [4X[32X[104X
  
  
  [1X66.11 [33X[0;0YPolynomials over the Rationals[133X[101X
  
  [33X[0;0YThe  following  functions  are  only  available to polynomials with rational
  coefficients:[133X
  
  [1X66.11-1 PrimitivePolynomial[101X
  
  [29X[2XPrimitivePolynomial[102X( [3Xf[103X ) [32X function
  
  [33X[0;0Ytakes  a  polynomial  [3Xf[103X  with  rational  coefficients  and  computes  a  new
  polynomial  with integral coefficients, obtained by multiplying with the Lcm
  of the denominators of the coefficients and casting out the content (the Gcd
  of  the  coefficients).  The  operation  returns  a list [[3Xnewpol[103X,[3Xcoeff[103X] with
  rational [3Xcoeff[103X such that [10X[3Xcoeff[103X[10X*[3Xnewpol[103X[10X=[3Xf[103X[10X[110X.[133X
  
  [1X66.11-2 PolynomialModP[101X
  
  [29X[2XPolynomialModP[102X( [3Xpol[103X, [3Xp[103X ) [32X function
  
  [33X[0;0Yfor  a  rational  polynomial [3Xpol[103X this function returns a polynomial over the
  field with [3Xp[103X elements, obtained by reducing the coefficients modulo [3Xp[103X.[133X
  
  [1X66.11-3 GaloisType[101X
  
  [29X[2XGaloisType[102X( [3Xf[103X[, [3Xcand[103X] ) [32X function
  
  [33X[0;0YLet [3Xf[103X be an irreducible polynomial with rational coefficients. This function
  returns  the type of Gal([3Xf[103X) (considered as a transitive permutation group of
  the  roots  of [3Xf[103X). It returns a number [3Xi[103X if Gal([3Xf[103X) is permutation isomorphic
  to [10XTransitiveGroup([3Xn[103X[10X,[3Xi[103X[10X)[110X where [3Xn[103X is the degree of [3Xf[103X.[133X
  
  [33X[0;0YIdentification  is  performed  by factoring appropriate Galois resolvents as
  proposed  in  [SM85].  This function is provided for rational polynomials of
  degree  up  to  15.  However, in some cases the required calculations become
  unfeasibly large.[133X
  
  [33X[0;0YFor  a  few  polynomials  of degree 14, a complete discrimination is not yet
  possible,  as  it  would  require  computations,  that are not feasible with
  current factoring methods.[133X
  
  [33X[0;0YThis  function  requires  the transitive groups library to be installed (see
  [14X50.6[114X).[133X
  
  [1X66.11-4 ProbabilityShapes[101X
  
  [29X[2XProbabilityShapes[102X( [3Xf[103X ) [32X function
  
  [33X[0;0YLet [3Xf[103X be an irreducible polynomial with rational coefficients. This function
  returns  a  list  of  the  most  likely  type(s)  of  Gal([3Xf[103X) (see [2XGaloisType[102X
  ([14X66.11-3[114X)),  based on factorization modulo a set of primes. It is very fast,
  but the result is only probabilistic.[133X
  
  [33X[0;0YThis  function  requires  the transitive groups library to be installed (see
  [14X50.6[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf:=x^9-9*x^7+27*x^5-39*x^3+36*x-8;;[127X[104X
    [4X[25Xgap>[125X [27XGaloisType(f);[127X[104X
    [4X[28X25[128X[104X
    [4X[25Xgap>[125X [27XTransitiveGroup(9,25);[127X[104X
    [4X[28X[1/2.S(3)^3]3[128X[104X
    [4X[25Xgap>[125X [27XProbabilityShapes(f);[127X[104X
    [4X[28X[ 25 ][128X[104X
  [4X[32X[104X
  
  
  [1X66.12 [33X[0;0YFactorization of Polynomials over the Rationals[133X[101X
  
  [33X[0;0YThe  following operations are used by [5XGAP[105X inside the factorization algorithm
  but might be of interest also in other contexts.[133X
  
  [1X66.12-1 BombieriNorm[101X
  
  [29X[2XBombieriNorm[102X( [3Xpol[103X ) [32X function
  
  [33X[0;0Ycomputes  weighted  Norm  [[3Xpol[103X][22X_2[122X  of [3Xpol[103X which is a good measure for factor
  coefficients (see [BTW93]).[133X
  
  [1X66.12-2 MinimizedBombieriNorm[101X
  
  [29X[2XMinimizedBombieriNorm[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YThis  function applies linear Tschirnhaus transformations ([22Xx ↦ x + i[122X) to the
  polynomial  [3Xf[103X, trying to get the Bombieri norm of [3Xf[103X small. It returns a list
  [10X[[3Xnew_polynomial[103X[10X, [3Xi_of_transformation[103X[10X][110X.[133X
  
  [1X66.12-3 HenselBound[101X
  
  [29X[2XHenselBound[102X( [3Xpol[103X[, [3Xminpol[103X, [3Xden[103X] ) [32X function
  
  [33X[0;0Yreturns  the  Hensel  bound  of the polynomial [3Xpol[103X. If the computation takes
  place  over  an  algebraic extension, then the minimal polynomial [3Xminpol[103X and
  denominator [3Xden[103X must be given.[133X
  
  [1X66.12-4 OneFactorBound[101X
  
  [29X[2XOneFactorBound[102X( [3Xpol[103X ) [32X function
  
  [33X[0;0Yreturns the coefficient bound for a single factor of the rational polynomial
  [3Xpol[103X.[133X
  
  
  [1X66.13 [33X[0;0YLaurent Polynomials[133X[101X
  
  [33X[0;0YA  univariate  polynomial  can  be written in the form [22Xr_0 + r_1 x + ⋯ + r_n
  x^n[122X,  with [22Xr_i ∈ R[122X. Formally, there is no reason to start with 0, if [22Xm[122X is an
  integer, we can consider objects of the form [22Xr_m x^m + r_{m+1} x^{m+1} + ⋯ +
  r_n  x^n[122X. We call these [13XLaurent polynomials[113X. Laurent polynomials also can be
  considered  as  quotients  of  a  univariate  polynomial  by  a power of the
  indeterminate.  The  addition  and  multiplication of univariate polynomials
  extends to Laurent polynomials (though it might be impossible to interpret a
  Laurent  polynomial  as  a  function)  and  many  functions  for  univariate
  polynomials  extend to Laurent polynomials (or extended versions for Laurent
  polynomials exist).[133X
  
  [1X66.13-1 LaurentPolynomialByCoefficients[101X
  
  [29X[2XLaurentPolynomialByCoefficients[102X( [3Xfam[103X, [3Xcofs[103X, [3Xval[103X[, [3Xind[103X] ) [32X operation
  
  [33X[0;0Yconstructs  a Laurent polynomial over the coefficients family [3Xfam[103X and in the
  indeterminate [3Xind[103X (defaulting to 1) with the coefficients given by [3Xcoefs[103X and
  valuation [3Xval[103X.[133X
  
  [1X66.13-2 CoefficientsOfLaurentPolynomial[101X
  
  [29X[2XCoefficientsOfLaurentPolynomial[102X( [3Xlaurent[103X ) [32X attribute
  
  [33X[0;0YFor  a  Laurent polynomial [3Xlaurent[103X, this function returns a pair [10X[[3Xcof[103X[10X, [3Xval[103X[10X][110X,
  consisting  of  the  coefficient  list  (in  ascending  order)  [3Xcof[103X  and the
  valuation [3Xval[103X of [3Xlaurent[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xp:=LaurentPolynomialByCoefficients(FamilyObj(1),[127X[104X
    [4X[25X>[125X [27X[1,2,3,4,5],-2);[127X[104X
    [4X[28X5*x^2+4*x+3+2*x^-1+x^-2[128X[104X
    [4X[25Xgap>[125X [27XNumeratorOfRationalFunction(p);DenominatorOfRationalFunction(p);[127X[104X
    [4X[28X5*x^4+4*x^3+3*x^2+2*x+1[128X[104X
    [4X[28Xx^2[128X[104X
    [4X[25Xgap>[125X [27XCoefficientsOfLaurentPolynomial(p*p);[127X[104X
    [4X[28X[ [ 1, 4, 10, 20, 35, 44, 46, 40, 25 ], -4 ][128X[104X
  [4X[32X[104X
  
  [1X66.13-3 IndeterminateNumberOfLaurentPolynomial[101X
  
  [29X[2XIndeterminateNumberOfLaurentPolynomial[102X( [3Xpol[103X ) [32X function
  
  [33X[0;0YIs a synonym for [2XIndeterminateNumberOfUnivariateRationalFunction[102X ([14X66.1-2[114X).[133X
  
  
  [1X66.14 [33X[0;0YUnivariate Rational Functions[133X[101X
  
  [1X66.14-1 UnivariateRationalFunctionByCoefficients[101X
  
  [29X[2XUnivariateRationalFunctionByCoefficients[102X( [3Xfam[103X, [3Xncof[103X, [3Xdcof[103X, [3Xval[103X[, [3Xind[103X] ) [32X operation
  
  [33X[0;0Yconstructs  a  univariate rational function over the coefficients family [3Xfam[103X
  and   in  the  indeterminate  [3Xind[103X  (defaulting  to  1)  with  numerator  and
  denominator coefficients given by [3Xncof[103X and [3Xdcof[103X and valuation [3Xval[103X.[133X
  
  
  [1X66.15 [33X[0;0YPolynomial Rings and Function Fields[133X[101X
  
  [33X[0;0YWhile  polynomials depend only on the family of the coefficients, polynomial
  rings  [22XA[122X  are defined over a base ring [22XR[122X. A polynomial is an element of [22XA[122X if
  and  only  if  all  its  coefficients  are contained in [22XR[122X. Besides providing
  domains  and  an easy way to create polynomials, polynomial rings can affect
  the behavior of operations like factorization into irreducibles.[133X
  
  [33X[0;0YIf  you  need  to  work  with  a  polynomial ring and its indeterminates the
  following  two  approaches will produce a ring that contains given variables
  (see  section [14X66.1[114X  for details about the internal numbering): Either, first
  create    the    ring    and    then    get    the    indeterminates    with
  [2XIndeterminatesOfPolynomialRing[102X ([14X66.15-2[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xr := PolynomialRing(Rationals,["a","b"]);;[127X[104X
    [4X[25Xgap>[125X [27Xindets := IndeterminatesOfPolynomialRing(r);;[127X[104X
    [4X[25Xgap>[125X [27Xa := indets[1]; a := indets[2];[127X[104X
    [4X[28Xa[128X[104X
    [4X[28Xb[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlternatively,  first  create  the  indeterminates  and then create the ring
  including these indeterminates.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:=Indeterminate(Rationals,"a":old);;[127X[104X
    [4X[25Xgap>[125X [27Xb:=Indeterminate(Rationals,"b":old);;[127X[104X
    [4X[25Xgap>[125X [27XPolynomialRing(Rationals,[a,b]);;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YAs  a convenient shortcut, intended mainly for interactive working, the [22Xi[122X-th
  indeterminate  of  a  polynomial  ring  [22XR[122X  can  be  accessed  as  [22XR.i[122X, which
  corresponds  exactly to [10XIndeterminatesOfPolynomialRing[110X[22X( R )[i][122X or, if it has
  the  name  [10Xnam[110X,  as  [22XR[122X[10X.nam[110X.  [13XNote[113X  that  the  number [22Xi[122X is in general [13Xnot[113X the
  indeterminate number, but simply an index into the indeterminates list of [22XR[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xr := PolynomialRing(Rationals, ["a", "b"]:old );;[127X[104X
    [4X[25Xgap>[125X [27Xr.1; r.2; r.a; r.b;[127X[104X
    [4X[28Xa[128X[104X
    [4X[28Xb[128X[104X
    [4X[28Xa[128X[104X
    [4X[28Xb[128X[104X
    [4X[25Xgap>[125X [27XIndeterminateNumberOfLaurentPolynomial(r.1);[127X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
  [33X[0;0YPolynomials as [5XGAP[105X objects can exist without a polynomial ring being defined
  and  polynomials  cannot be associated to a particular polynomial ring. (For
  example  dividing  a  polynomial  which  is  in  a  polynomial ring over the
  integers  by another integer will result in a polynomial over the rationals,
  not in a rational function over the integers.)[133X
  
  
  [1X66.15-1 [33X[0;0YPolynomialRing[133X[101X
  
  [29X[2XPolynomialRing[102X( [3XR[103X, [3Xrank[103X[, [3Xavoid[103X] ) [32X operation
  [29X[2XPolynomialRing[102X( [3XR[103X, [3Xnames[103X[, [3Xavoid[103X] ) [32X operation
  [29X[2XPolynomialRing[102X( [3XR[103X, [3Xindets[103X ) [32X operation
  [29X[2XPolynomialRing[102X( [3XR[103X, [3Xindetnums[103X ) [32X operation
  
  [33X[0;0Ycreates  a  polynomial  ring  over the ring [3XR[103X. If a positive integer [3Xrank[103X is
  given,  this  creates  the  polynomial  ring  in  [3Xrank[103X indeterminates. These
  indeterminates  will  have  the internal index numbers 1 to [3Xrank[103X. The second
  usage  takes  a  list  [3Xnames[103X  of  strings  and  returns a polynomial ring in
  indeterminates  labelled  by  [3Xnames[103X.  These indeterminates have [21Xnew[121X internal
  index  numbers  as  if  they  had  been  created  by  calls to [2XIndeterminate[102X
  ([14X66.1-1[114X).  (If  the  argument [3Xavoid[103X is given it contains indeterminates that
  should  be  avoided,  in this case internal index numbers are incremented to
  skip these variables.) In the third version, a list of indeterminates [3Xindets[103X
  is  given.  This  creates  the polynomial ring in the indeterminates [3Xindets[103X.
  Finally, the fourth version specifies indeterminates by their index numbers.[133X
  
  [33X[0;0YTo     get     the    indeterminates    of    a    polynomial    ring    use
  [2XIndeterminatesOfPolynomialRing[102X     ([14X66.15-2[114X).     (Indeterminates    created
  independently  with  [2XIndeterminate[102X ([14X66.1-1[114X) will usually differ, though they
  might be given the same name and display identically, see Section [14X66.1[114X.)[133X
  
  [1X66.15-2 IndeterminatesOfPolynomialRing[101X
  
  [29X[2XIndeterminatesOfPolynomialRing[102X( [3Xpring[103X ) [32X attribute
  [29X[2XIndeterminatesOfFunctionField[102X( [3Xffield[103X ) [32X attribute
  
  [33X[0;0Yreturns  a  list  of  the  indeterminates  of  the  polynomial  ring  [3Xpring[103X,
  respectively the function field [3Xffield[103X.[133X
  
  [1X66.15-3 CoefficientsRing[101X
  
  [29X[2XCoefficientsRing[102X( [3Xpring[103X ) [32X attribute
  
  [33X[0;0Yreturns  the  ring of coefficients of the polynomial ring [3Xpring[103X, that is the
  ring over which [3Xpring[103X was defined.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xr:=PolynomialRing(GF(7));[127X[104X
    [4X[28XGF(7)[x_1][128X[104X
    [4X[25Xgap>[125X [27Xr:=PolynomialRing(GF(7),3);[127X[104X
    [4X[28XGF(7)[x_1,x_2,x_3][128X[104X
    [4X[25Xgap>[125X [27XIndeterminatesOfPolynomialRing(r);[127X[104X
    [4X[28X[ x_1, x_2, x_3 ][128X[104X
    [4X[25Xgap>[125X [27Xr2:=PolynomialRing(GF(7),[5,7,12]);[127X[104X
    [4X[28XGF(7)[x_5,x_7,x_12][128X[104X
    [4X[25Xgap>[125X [27XCoefficientsRing(r);[127X[104X
    [4X[28XGF(7)[128X[104X
    [4X[25Xgap>[125X [27Xr:=PolynomialRing(GF(7),3);[127X[104X
    [4X[28XGF(7)[x_1,x_2,x_3][128X[104X
    [4X[25Xgap>[125X [27Xr2:=PolynomialRing(GF(7),3,IndeterminatesOfPolynomialRing(r));[127X[104X
    [4X[28XGF(7)[x_4,x_5,x_6][128X[104X
    [4X[25Xgap>[125X [27Xr:=PolynomialRing(GF(7),["x","y","z","z2"]);[127X[104X
    [4X[28XGF(7)[x,y,z,z2][128X[104X
  [4X[32X[104X
  
  [1X66.15-4 IsPolynomialRing[101X
  
  [29X[2XIsPolynomialRing[102X( [3Xpring[103X ) [32X Category
  
  [33X[0;0Yis the category of polynomial rings[133X
  
  [1X66.15-5 IsFiniteFieldPolynomialRing[101X
  
  [29X[2XIsFiniteFieldPolynomialRing[102X( [3Xpring[103X ) [32X Category
  
  [33X[0;0Yis the category of polynomial rings over a finite field (see Chapter [14X59[114X).[133X
  
  [1X66.15-6 IsAbelianNumberFieldPolynomialRing[101X
  
  [29X[2XIsAbelianNumberFieldPolynomialRing[102X( [3Xpring[103X ) [32X Category
  
  [33X[0;0Yis  the  category  of  polynomial rings over a field of cyclotomics (see the
  chapters [14X18[114X and [14X60[114X).[133X
  
  [1X66.15-7 IsRationalsPolynomialRing[101X
  
  [29X[2XIsRationalsPolynomialRing[102X( [3Xpring[103X ) [32X Category
  
  [33X[0;0Yis the category of polynomial rings over the rationals (see Chapter [14X17[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xr := PolynomialRing(Rationals, ["a", "b"] );;[127X[104X
    [4X[25Xgap>[125X [27XIsPolynomialRing(r);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsFiniteFieldPolynomialRing(r);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsRationalsPolynomialRing(r);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X66.15-8 [33X[0;0YFunctionField[133X[101X
  
  [29X[2XFunctionField[102X( [3XR[103X, [3Xrank[103X[, [3Xavoid[103X] ) [32X operation
  [29X[2XFunctionField[102X( [3XR[103X, [3Xnames[103X[, [3Xavoid[103X] ) [32X operation
  [29X[2XFunctionField[102X( [3XR[103X, [3Xindets[103X ) [32X operation
  [29X[2XFunctionField[102X( [3XR[103X, [3Xindetnums[103X ) [32X operation
  
  [33X[0;0Ycreates  a  function  field  over the integral ring [3XR[103X. If a positive integer
  [3Xrank[103X is given, this creates the function field in [3Xrank[103X indeterminates. These
  indeterminates  will  have  the internal index numbers 1 to [3Xrank[103X. The second
  usage  takes  a  list  [3Xnames[103X  of  strings  and  returns  a function field in
  indeterminates  labelled  by  [3Xnames[103X.  These indeterminates have [21Xnew[121X internal
  index  numbers  as  if  they  had  been  created  by  calls to [2XIndeterminate[102X
  ([14X66.1-1[114X).  (If  the  argument [3Xavoid[103X is given it contains indeterminates that
  should  be  avoided,  in this case internal index numbers are incremented to
  skip these variables.) In the third version, a list of indeterminates [3Xindets[103X
  is  given.  This  creates  the  function field in the indeterminates [3Xindets[103X.
  Finally, the fourth version specifies indeterminates by their index number.[133X
  
  [33X[0;0YTo     get     the     indeterminates    of    a    function    field    use
  [2XIndeterminatesOfFunctionField[102X     ([14X66.15-2[114X).     (Indeterminates     created
  independently  with  [2XIndeterminate[102X ([14X66.1-1[114X) will usually differ, though they
  might be given the same name and display identically, see Section [14X66.1[114X.)[133X
  
  [1X66.15-9 IsFunctionField[101X
  
  [29X[2XIsFunctionField[102X( [3Xffield[103X ) [32X Category
  
  [33X[0;0Yis the category of function fields[133X
  
  
  [1X66.16 [33X[0;0YUnivariate Polynomial Rings[133X[101X
  
  
  [1X66.16-1 [33X[0;0YUnivariatePolynomialRing[133X[101X
  
  [29X[2XUnivariatePolynomialRing[102X( [3XR[103X[, [3Xnr[103X] ) [32X operation
  [29X[2XUnivariatePolynomialRing[102X( [3XR[103X[, [3Xname[103X][, [3Xavoid[103X] ) [32X operation
  
  [33X[0;0Yreturns  a  univariate polynomial ring in the indeterminate [3Xnr[103X over the base
  ring [3XR[103X. If [3Xnr[103X is not given it defaults to 1.[133X
  
  [33X[0;0YIf  the number is not specified a list [3Xavoid[103X of indeterminates may be given.
  Then  the function will return a ring in an indeterminate that is guaranteed
  to be different from all the indeterminates in [3Xavoid[103X.[133X
  
  [33X[0;0YAlso a string [3Xname[103X can be prescribed as the name of the indeterminate chosen
  (also avoiding the indeterminates in the list [3Xavoid[103X if given).[133X
  
  [1X66.16-2 IsUnivariatePolynomialRing[101X
  
  [29X[2XIsUnivariatePolynomialRing[102X( [3Xpring[103X ) [32X Category
  
  [33X[0;0Yis the category of polynomial rings with one indeterminate.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xr:=UnivariatePolynomialRing(Rationals,"p");[127X[104X
    [4X[28XRationals[p][128X[104X
    [4X[25Xgap>[125X [27Xr2:=PolynomialRing(Rationals,["q"]);[127X[104X
    [4X[28XRationals[q][128X[104X
    [4X[25Xgap>[125X [27XIsUnivariatePolynomialRing(r);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsUnivariatePolynomialRing(r2);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X66.17 [33X[0;0YMonomial Orderings[133X[101X
  
  [33X[0;0YIt  is  often  desirable to consider the monomials within a polynomial to be
  arranged  with  respect  to a certain ordering. Such an ordering is called a
  [13Xmonomial  ordering[113X if it is total, invariant under multiplication with other
  monomials  and admits no infinite descending chains. For details on monomial
  orderings see [CLO97].[133X
  
  [33X[0;0YIn  [5XGAP[105X, monomial orderings are represented by objects that provide a way to
  compare monomials (as polynomials as well as –for efficiency purposes within
  algorithms– in the internal representation as lists).[133X
  
  [33X[0;0YNormally  the  ordering  chosen  should  be  [13Xadmissible[113X,  i.e.  it  must  be
  compatible  with  products: If [22Xa < b[122X then [22Xca < cb[122X for all monomials [22Xa, b[122X and
  [22Xc[122X.[133X
  
  [33X[0;0YEach monomial ordering provides the two functions [2XMonomialComparisonFunction[102X
  ([14X66.17-5[114X)  and  [2XMonomialExtrepComparisonFun[102X  ([14X66.17-6[114X) to compare monomials.
  These  functions  work as [21Xis less than[121X, i.e. they return [9Xtrue[109X if and only if
  the left argument is smaller.[133X
  
  [1X66.17-1 IsMonomialOrdering[101X
  
  [29X[2XIsMonomialOrdering[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YA  monomial  ordering  is  an  object  representing a monomial ordering. Its
  attributes          [2XMonomialComparisonFunction[102X         ([14X66.17-5[114X)         and
  [2XMonomialExtrepComparisonFun[102X ([14X66.17-6[114X) are actual comparison functions.[133X
  
  [1X66.17-2 LeadingMonomialOfPolynomial[101X
  
  [29X[2XLeadingMonomialOfPolynomial[102X( [3Xpol[103X, [3Xord[103X ) [32X function
  
  [33X[0;0Yreturns  the  leading  monomial  (with  respect  to the ordering [3Xord[103X) of the
  polynomial [3Xpol[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xx:=Indeterminate(Rationals,"x");;[127X[104X
    [4X[25Xgap>[125X [27Xy:=Indeterminate(Rationals,"y");;[127X[104X
    [4X[25Xgap>[125X [27Xz:=Indeterminate(Rationals,"z");;[127X[104X
    [4X[25Xgap>[125X [27Xlexord:=MonomialLexOrdering();grlexord:=MonomialGrlexOrdering();[127X[104X
    [4X[28XMonomialLexOrdering()[128X[104X
    [4X[28XMonomialGrlexOrdering()[128X[104X
    [4X[25Xgap>[125X [27Xf:=2*x+3*y+4*z+5*x^2-6*z^2+7*y^3;          [127X[104X
    [4X[28X7*y^3+5*x^2-6*z^2+2*x+3*y+4*z[128X[104X
    [4X[25Xgap>[125X [27XLeadingMonomialOfPolynomial(f,lexord);[127X[104X
    [4X[28Xx^2[128X[104X
    [4X[25Xgap>[125X [27XLeadingMonomialOfPolynomial(f,grlexord);[127X[104X
    [4X[28Xy^3[128X[104X
  [4X[32X[104X
  
  [1X66.17-3 LeadingTermOfPolynomial[101X
  
  [29X[2XLeadingTermOfPolynomial[102X( [3Xpol[103X, [3Xord[103X ) [32X function
  
  [33X[0;0Yreturns  the  leading  term  (with  respect  to  the  ordering  [3Xord[103X)  of the
  polynomial  [3Xpol[103X,  i.e.  the  product  of  leading  coefficient  and  leading
  monomial.[133X
  
  [1X66.17-4 LeadingCoefficientOfPolynomial[101X
  
  [29X[2XLeadingCoefficientOfPolynomial[102X( [3Xpol[103X, [3Xord[103X ) [32X operation
  
  [33X[0;0Yreturns  the  leading  coefficient  (that  is the coefficient of the leading
  monomial, see [2XLeadingMonomialOfPolynomial[102X ([14X66.17-2[114X)) of the polynomial [3Xpol[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLeadingTermOfPolynomial(f,lexord);[127X[104X
    [4X[28X5*x^2[128X[104X
    [4X[25Xgap>[125X [27XLeadingTermOfPolynomial(f,grlexord);[127X[104X
    [4X[28X7*y^3[128X[104X
    [4X[25Xgap>[125X [27XLeadingCoefficientOfPolynomial(f,lexord);[127X[104X
    [4X[28X5[128X[104X
  [4X[32X[104X
  
  [1X66.17-5 MonomialComparisonFunction[101X
  
  [29X[2XMonomialComparisonFunction[102X( [3XO[103X ) [32X attribute
  
  [33X[0;0YIf [3XO[103X is an object representing a monomial ordering, this attribute returns a
  [13Xfunction[113X  that  can  be  used  to compare or sort monomials (and polynomials
  which  will  be  compared  by  their  monomials in decreasing order) in this
  order.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMonomialComparisonFunction(lexord);[127X[104X
    [4X[28Xfunction( a, b ) ... end[128X[104X
    [4X[25Xgap>[125X [27Xl:=[f,Derivative(f,x),Derivative(f,y),Derivative(f,z)];;[127X[104X
    [4X[25Xgap>[125X [27XSort(l,MonomialComparisonFunction(lexord));l;[127X[104X
    [4X[28X[ -12*z+4, 21*y^2+3, 10*x+2, 7*y^3+5*x^2-6*z^2+2*x+3*y+4*z ][128X[104X
  [4X[32X[104X
  
  [1X66.17-6 MonomialExtrepComparisonFun[101X
  
  [29X[2XMonomialExtrepComparisonFun[102X( [3XO[103X ) [32X attribute
  
  [33X[0;0YIf [3XO[103X is an object representing a monomial ordering, this attribute returns a
  [13Xfunction[113X  that  can  be  used to compare or sort monomials [13Xin their external
  representation[113X (as lists). This comparison variant is used inside algorithms
  that manipulate the external representation.[133X
  
  [1X66.17-7 MonomialLexOrdering[101X
  
  [29X[2XMonomialLexOrdering[102X( [[3Xvari[103X] ) [32X function
  
  [33X[0;0YThis  function creates a lexicographic ordering for monomials. Monomials are
  compared  first by the exponents of the largest variable, then the exponents
  of the second largest variable and so on.[133X
  
  [33X[0;0YThe  variables are ordered according to their (internal) index, i.e., [22Xx_1[122X is
  larger  than  [22Xx_2[122X and so on. If [3Xvari[103X is given, and is a list of variables or
  variable  indices,  instead  this  arrangement  of  variables (in descending
  order;  i.e.  the  first  variable is larger than the second) is used as the
  underlying order of variables.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xl:=List(Tuples([1..3],3),i->x^(i[1]-1)*y^(i[2]-1)*z^(i[3]-1));[127X[104X
    [4X[28X[ 1, z, z^2, y, y*z, y*z^2, y^2, y^2*z, y^2*z^2, x, x*z, x*z^2, x*y, [128X[104X
    [4X[28X  x*y*z, x*y*z^2, x*y^2, x*y^2*z, x*y^2*z^2, x^2, x^2*z, x^2*z^2, [128X[104X
    [4X[28X  x^2*y, x^2*y*z, x^2*y*z^2, x^2*y^2, x^2*y^2*z, x^2*y^2*z^2 ][128X[104X
    [4X[25Xgap>[125X [27XSort(l,MonomialComparisonFunction(MonomialLexOrdering()));l;[127X[104X
    [4X[28X[ 1, z, z^2, y, y*z, y*z^2, y^2, y^2*z, y^2*z^2, x, x*z, x*z^2, x*y, [128X[104X
    [4X[28X  x*y*z, x*y*z^2, x*y^2, x*y^2*z, x*y^2*z^2, x^2, x^2*z, x^2*z^2, [128X[104X
    [4X[28X  x^2*y, x^2*y*z, x^2*y*z^2, x^2*y^2, x^2*y^2*z, x^2*y^2*z^2 ][128X[104X
    [4X[25Xgap>[125X [27XSort(l,MonomialComparisonFunction(MonomialLexOrdering([y,z,x])));l;[127X[104X
    [4X[28X[ 1, x, x^2, z, x*z, x^2*z, z^2, x*z^2, x^2*z^2, y, x*y, x^2*y, y*z, [128X[104X
    [4X[28X  x*y*z, x^2*y*z, y*z^2, x*y*z^2, x^2*y*z^2, y^2, x*y^2, x^2*y^2, [128X[104X
    [4X[28X  y^2*z, x*y^2*z, x^2*y^2*z, y^2*z^2, x*y^2*z^2, x^2*y^2*z^2 ][128X[104X
    [4X[25Xgap>[125X [27XSort(l,MonomialComparisonFunction(MonomialLexOrdering([z,x,y])));l;[127X[104X
    [4X[28X[ 1, y, y^2, x, x*y, x*y^2, x^2, x^2*y, x^2*y^2, z, y*z, y^2*z, x*z, [128X[104X
    [4X[28X  x*y*z, x*y^2*z, x^2*z, x^2*y*z, x^2*y^2*z, z^2, y*z^2, y^2*z^2, [128X[104X
    [4X[28X  x*z^2, x*y*z^2, x*y^2*z^2, x^2*z^2, x^2*y*z^2, x^2*y^2*z^2 ][128X[104X
  [4X[32X[104X
  
  [1X66.17-8 MonomialGrlexOrdering[101X
  
  [29X[2XMonomialGrlexOrdering[102X( [[3Xvari[103X] ) [32X function
  
  [33X[0;0YThis  function  creates  a  degree/lexicographic  ordering. In this ordering
  monomials  are  compared first by their total degree, then lexicographically
  (see [2XMonomialLexOrdering[102X ([14X66.17-7[114X)).[133X
  
  [33X[0;0YThe  variables are ordered according to their (internal) index, i.e., [22Xx_1[122X is
  larger  than  [22Xx_2[122X and so on. If [3Xvari[103X is given, and is a list of variables or
  variable  indices,  instead  this  arrangement  of  variables (in descending
  order;  i.e.  the  first  variable is larger than the second) is used as the
  underlying order of variables.[133X
  
  [1X66.17-9 MonomialGrevlexOrdering[101X
  
  [29X[2XMonomialGrevlexOrdering[102X( [[3Xvari[103X] ) [32X function
  
  [33X[0;0YThis  function  creates  a  [21Xgrevlex[121X ordering. In this ordering monomials are
  compared  first  by total degree and then backwards lexicographically. (This
  is different than [21Xgrlex[121X ordering with variables reversed.)[133X
  
  [33X[0;0YThe  variables are ordered according to their (internal) index, i.e., [22Xx_1[122X is
  larger  than  [22Xx_2[122X and so on. If [3Xvari[103X is given, and is a list of variables or
  variable  indices,  instead  this  arrangement  of  variables (in descending
  order;  i.e.  the  first  variable is larger than the second) is used as the
  underlying order of variables.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSort(l,MonomialComparisonFunction(MonomialGrlexOrdering()));l;[127X[104X
    [4X[28X[ 1, z, y, x, z^2, y*z, y^2, x*z, x*y, x^2, y*z^2, y^2*z, x*z^2, [128X[104X
    [4X[28X  x*y*z, x*y^2, x^2*z, x^2*y, y^2*z^2, x*y*z^2, x*y^2*z, x^2*z^2, [128X[104X
    [4X[28X  x^2*y*z, x^2*y^2, x*y^2*z^2, x^2*y*z^2, x^2*y^2*z, x^2*y^2*z^2 ][128X[104X
    [4X[25Xgap>[125X [27XSort(l,MonomialComparisonFunction(MonomialGrevlexOrdering()));l;[127X[104X
    [4X[28X[ 1, z, y, x, z^2, y*z, x*z, y^2, x*y, x^2, y*z^2, x*z^2, y^2*z, [128X[104X
    [4X[28X  x*y*z, x^2*z, x*y^2, x^2*y, y^2*z^2, x*y*z^2, x^2*z^2, x*y^2*z, [128X[104X
    [4X[28X  x^2*y*z, x^2*y^2, x*y^2*z^2, x^2*y*z^2, x^2*y^2*z, x^2*y^2*z^2 ][128X[104X
    [4X[25Xgap>[125X [27XSort(l,MonomialComparisonFunction(MonomialGrlexOrdering([z,y,x])));l;[127X[104X
    [4X[28X[ 1, x, y, z, x^2, x*y, y^2, x*z, y*z, z^2, x^2*y, x*y^2, x^2*z, [128X[104X
    [4X[28X  x*y*z, y^2*z, x*z^2, y*z^2, x^2*y^2, x^2*y*z, x*y^2*z, x^2*z^2, [128X[104X
    [4X[28X  x*y*z^2, y^2*z^2, x^2*y^2*z, x^2*y*z^2, x*y^2*z^2, x^2*y^2*z^2 ][128X[104X
  [4X[32X[104X
  
  [1X66.17-10 EliminationOrdering[101X
  
  [29X[2XEliminationOrdering[102X( [3Xelim[103X[, [3Xrest[103X] ) [32X function
  
  [33X[0;0YThis  function creates an elimination ordering for eliminating the variables
  in  [3Xelim[103X.  Two  monomials are compared first by the exponent vectors for the
  variables  listed  in  [3Xelim[103X  (a lexicographic comparison with respect to the
  ordering   indicated   in   [3Xelim[103X).  If  these  submonomial  are  equal,  the
  submonomials  given  by  the  other  variables  are  compared  by  a  graded
  lexicographic ordering (with respect to the variable order given in [3Xrest[103X, if
  called with two parameters).[133X
  
  [33X[0;0YBoth [3Xelim[103X and [3Xrest[103X may be a list of variables or a list of variable indices.[133X
  
  [1X66.17-11 PolynomialReduction[101X
  
  [29X[2XPolynomialReduction[102X( [3Xpoly[103X, [3Xgens[103X, [3Xorder[103X ) [32X function
  
  [33X[0;0Yreduces  the  polynomial  [3Xpoly[103X  by the ideal generated by the polynomials in
  [3Xgens[103X, using the order [3Xorder[103X of monomials. Unless [3Xgens[103X is a Gröbner basis the
  result is not guaranteed to be unique.[133X
  
  [33X[0;0YThe operation returns a list of length two, the first entry is the remainder
  after  the  reduction. The second entry is a list of quotients corresponding
  to [3Xgens[103X.[133X
  
  [33X[0;0YNote that the strategy used by [2XPolynomialReduction[102X differs from the standard
  textbook      reduction      algorithm,     which     is     provided     by
  [2XPolynomialDivisionAlgorithm[102X ([14X66.17-13[114X).[133X
  
  [1X66.17-12 PolynomialReducedRemainder[101X
  
  [29X[2XPolynomialReducedRemainder[102X( [3Xpoly[103X, [3Xgens[103X, [3Xorder[103X ) [32X function
  
  [33X[0;0Ythis  operation does the same way as [2XPolynomialReduction[102X ([14X66.17-11[114X) but does
  not keep track of the actual quotients and returns only the remainder (it is
  therefore slightly faster).[133X
  
  [1X66.17-13 PolynomialDivisionAlgorithm[101X
  
  [29X[2XPolynomialDivisionAlgorithm[102X( [3Xpoly[103X, [3Xgens[103X, [3Xorder[103X ) [32X function
  
  [33X[0;0YThis function implements the division algorithm for multivariate polynomials
  as  given  in  [CLO97,  Theorem  3  in  Chapter 2]. (It might be slower than
  [2XPolynomialReduction[102X  ([14X66.17-11[114X)  but  the remainders are guaranteed to agree
  with the textbook.)[133X
  
  [33X[0;0YThe operation returns a list of length two, the first entry is the remainder
  after  the  reduction. The second entry is a list of quotients corresponding
  to [3Xgens[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbas:=[x^3*y*z,x*y^2*z,z*y*z^3+x];;[127X[104X
    [4X[25Xgap>[125X [27Xpol:=x^7*z*bas[1]+y^5*bas[3]+x*z;;[127X[104X
    [4X[25Xgap>[125X [27XPolynomialReduction(pol,bas,MonomialLexOrdering()); [127X[104X
    [4X[28X[ -y*z^5, [ x^7*z, 0, y^5+z ] ][128X[104X
    [4X[25Xgap>[125X [27XPolynomialReducedRemainder(pol,bas,MonomialLexOrdering());[127X[104X
    [4X[28X-y*z^5[128X[104X
    [4X[25Xgap>[125X [27XPolynomialDivisionAlgorithm(pol,bas,MonomialLexOrdering());[127X[104X
    [4X[28X[ -y*z^5, [ x^7*z, 0, y^5+z ] ][128X[104X
  [4X[32X[104X
  
  [1X66.17-14 MonomialExtGrlexLess[101X
  
  [29X[2XMonomialExtGrlexLess[102X( [3Xa[103X, [3Xb[103X ) [32X function
  
  [33X[0;0Yimplements  comparison  of  monomial  in  their external representation by a
  [21Xgrlex[121X  order  with  [22Xx_1>x_2[122X  (This  is  exactly  the same as the ordering by
  [2XMonomialGrlexOrdering[102X   ([14X66.17-8[114X),  see   [14X66.17[114X).  The  function  takes  two
  monomials  [3Xa[103X and [3Xb[103X in expanded form and returns whether the first is smaller
  than  the  second.  (This  ordering  is  also  used  by  [5XGAP[105X  internally for
  representing polynomials as a linear combination of monomials.)[133X
  
  [33X[0;0YSee section [14X66.21[114X for details on the expanded form of monomials.[133X
  
  
  [1X66.18 [33X[0;0YGroebner Bases[133X[101X
  
  [33X[0;0YA  [13XGroebner  Basis[113X of an ideal [22XI[122Xi, in a polynomial ring [22XR[122X, with respect to a
  monomial  ordering,  is  a  set  of  ideal  generators [3XG[103X such that the ideal
  generated  by  the leading monomials of all polynomials in [3XG[103X is equal to the
  ideal generated by the leading monomials of all polynomials in [3XI[103X.[133X
  
  [33X[0;0YFor more details on Groebner bases see [CLO97].[133X
  
  
  [1X66.18-1 [33X[0;0YGroebnerBasis[133X[101X
  
  [29X[2XGroebnerBasis[102X( [3XL[103X, [3XO[103X ) [32X operation
  [29X[2XGroebnerBasis[102X( [3XI[103X, [3XO[103X ) [32X operation
  [29X[2XGroebnerBasisNC[102X( [3XL[103X, [3XO[103X ) [32X operation
  
  [33X[0;0YLet [3XO[103X be a monomial ordering and [3XL[103X be a list of polynomials that generate an
  ideal  [3XI[103X.  This  operation returns a Groebner basis of [3XI[103X with respect to the
  ordering [3XO[103X.[133X
  
  [33X[0;0Y[2XGroebnerBasisNC[102X  works like [2XGroebnerBasis[102X with the only distinction that the
  first argument has to be a list of polynomials and that no test is performed
  to check whether the ordering is defined for all occuring variables.[133X
  
  [33X[0;0YNote  that  [5XGAP[105X  at  the  moment  only  includes  a  naïve implementation of
  Buchberger's  algorithm  (which  is  mainly intended as a teaching tool). It
  might not be sufficient for serious problems.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xl:=[x^2+y^2+z^2-1,x^2+z^2-y,x-y];;[127X[104X
    [4X[25Xgap>[125X [27XGroebnerBasis(l,MonomialLexOrdering());[127X[104X
    [4X[28X[ x^2+y^2+z^2-1, x^2+z^2-y, x-y, -y^2-y+1, -z^2+2*y-1, [128X[104X
    [4X[28X  1/2*z^4+2*z^2-1/2 ][128X[104X
    [4X[25Xgap>[125X [27XGroebnerBasis(l,MonomialLexOrdering([z,x,y]));[127X[104X
    [4X[28X[ x^2+y^2+z^2-1, x^2+z^2-y, x-y, -y^2-y+1 ][128X[104X
    [4X[25Xgap>[125X [27XGroebnerBasis(l,MonomialGrlexOrdering());[127X[104X
    [4X[28X[ x^2+y^2+z^2-1, x^2+z^2-y, x-y, -y^2-y+1, -z^2+2*y-1 ][128X[104X
  [4X[32X[104X
  
  
  [1X66.18-2 [33X[0;0YReducedGroebnerBasis[133X[101X
  
  [29X[2XReducedGroebnerBasis[102X( [3XL[103X, [3XO[103X ) [32X operation
  [29X[2XReducedGroebnerBasis[102X( [3XI[103X, [3XO[103X ) [32X operation
  
  [33X[0;0Ya  Groebner  basis [22XB[122X (see [2XGroebnerBasis[102X ([14X66.18-1[114X)) is [13Xreduced[113X if no monomial
  in  a  polynomial  in  [3XB[103X  is  divisible  by  the leading monomial of another
  polynomial  in  [22XB[122X.  This operation computes a Groebner basis with respect to
  the monomial ordering [3XO[103X and then reduces it.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XReducedGroebnerBasis(l,MonomialGrlexOrdering());[127X[104X
    [4X[28X[ x-y, z^2-2*y+1, y^2+y-1 ][128X[104X
    [4X[25Xgap>[125X [27XReducedGroebnerBasis(l,MonomialLexOrdering());    [127X[104X
    [4X[28X[ z^4+4*z^2-1, -1/2*z^2+y-1/2, -1/2*z^2+x-1/2 ][128X[104X
    [4X[25Xgap>[125X [27XReducedGroebnerBasis(l,MonomialLexOrdering([y,z,x]));[127X[104X
    [4X[28X[ x^2+x-1, z^2-2*x+1, -x+y ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  performance reasons it can be advantageous to define monomial orderings
  once and then to reuse them:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xord:=MonomialGrlexOrdering();;[127X[104X
    [4X[25Xgap>[125X [27XGroebnerBasis(l,ord);[127X[104X
    [4X[28X[ x^2+y^2+z^2-1, x^2+z^2-y, x-y, -y^2-y+1, -z^2+2*y-1 ][128X[104X
    [4X[25Xgap>[125X [27XReducedGroebnerBasis(l,ord);[127X[104X
    [4X[28X[ x-y, z^2-2*y+1, y^2+y-1 ][128X[104X
  [4X[32X[104X
  
  [1X66.18-3 StoredGroebnerBasis[101X
  
  [29X[2XStoredGroebnerBasis[102X( [3XI[103X ) [32X attribute
  
  [33X[0;0YFor  an  ideal  [3XI[103X in a polynomial ring, this attribute holds a list [22X[ B, O ][122X
  where [22XB[122X is a Groebner basis for the monomial ordering [22XO[122X. this can be used to
  test membership or canonical coset representatives.[133X
  
  [1X66.18-4 InfoGroebner[101X
  
  [29X[2XInfoGroebner[102X[32X info class
  
  [33X[0;0YThis info class gives information about Groebner basis calculations.[133X
  
  
  [1X66.19 [33X[0;0YRational Function Families[133X[101X
  
  [33X[0;0YAll  rational  functions  defined  over  a  ring lie in the same family, the
  rational functions family over this ring.[133X
  
  [33X[0;0YIn  [5XGAP[105X  therefore  the family of a polynomial depends only on the family of
  the  coefficients, all polynomials whose coefficients lie in the same family
  are [21Xcompatible[121X.[133X
  
  [1X66.19-1 RationalFunctionsFamily[101X
  
  [29X[2XRationalFunctionsFamily[102X( [3Xfam[103X ) [32X attribute
  
  [33X[0;0Ycreates a family containing rational functions with coefficients in [3Xfam[103X. All
  elements    of    the   [2XRationalFunctionsFamily[102X   are   rational   functions
  (see [2XIsRationalFunction[102X ([14X66.4-1[114X)).[133X
  
  [1X66.19-2 IsPolynomialFunctionsFamily[101X
  
  [29X[2XIsPolynomialFunctionsFamily[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsRationalFunctionsFamily[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0Y[2XIsPolynomialFunctionsFamily[102X  is the category of a family of polynomials. For
  families  over  an  UFD,  the category becomes [2XIsRationalFunctionsFamily[102X (as
  rational  functions  and  quotients  are  only provided for families over an
  UFD.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfam:=RationalFunctionsFamily(FamilyObj(1));[127X[104X
    [4X[28XNewFamily( "RationalFunctionsFamily(...)", [ 618, 620 ],[128X[104X
    [4X[28X[ 82, 85, 89, 93, 97, 100, 103, 107, 111, 618, 620 ] )[128X[104X
  [4X[32X[104X
  
  [1X66.19-3 CoefficientsFamily[101X
  
  [29X[2XCoefficientsFamily[102X( [3Xrffam[103X ) [32X attribute
  
  [33X[0;0YIf  [3Xrffam[103X  has  been created as [10XRationalFunctionsFamily([3Xcfam[103X[10X)[110X this attribute
  holds the coefficients family [3Xcfam[103X.[133X
  
  [33X[0;0Y[5XGAP[105X   does   [13Xnot[113X   embed  the  base  ring  in  the  polynomial  ring.  While
  multiplication  and  addition  of  base  ring elements to rational functions
  return  the  expected  results,  polynomials  and rational functions are not
  equal.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X1=Indeterminate(Rationals)^0;[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  
  [1X66.20 [33X[0;0YThe Representations of Rational Functions[133X[101X
  
  [33X[0;0Y[5XGAP[105X  uses  four  representations  of  rational functions: Rational functions
  given   by  numerator  and  denominator,  polynomials,  univariate  rational
  functions  (given  by  coefficient  lists  for numerator and denominator and
  valuation)   and   Laurent   polynomials  (given  by  coefficient  list  and
  valuation).[133X
  
  [33X[0;0YThese  representations  do  not necessarily reflect mathematical properties:
  While  an object in the Laurent polynomials representation must be a Laurent
  polynomial it might turn out that a rational function given by numerator and
  denominator  is  actually  a  Laurent  polynomial  and the property tests in
  section [14X66.4[114X will find this out.[133X
  
  [33X[0;0YEach  representation  is  associated one or several [21Xdefining attributes[121X that
  give  an  [21Xexternal[121X  representation  (see [14X66.21[114X) of the representation in the
  form  of  lists  and  are  the  defining  information  that tells a rational
  function what it is.[133X
  
  [33X[0;0Y[5XGAP[105X  also  implements  methods  to  compute  these  attributes  for rational
  functions in [13Xother[113X representations, provided it would be possible to express
  an  [13Xmathematically  equal[113X rational function in the representation associated
  with  the  attribute.  (That  is  one can always get a numerator/denominator
  representation  of  a  polynomial  while an arbitrary function of course can
  compute a polynomial representation only if it is a polynomial.)[133X
  
  [33X[0;0YTherefore  these  attributes can be thought of as [21Xconceptual[121X representations
  that  allow  us  –as  far  as  possible– to consider an object as a rational
  function,  a polynomial or a Laurent polynomial, regardless of the way it is
  represented in the computer.[133X
  
  [33X[0;0YFunctions  thus  usually  do  not need to care about the representation of a
  rational  function.  Depending  on  its (known in the context or determined)
  properties, they can access the attribute representing the rational function
  in the desired way.[133X
  
  [33X[0;0YConsequentially, methods for rational functions are installed for properties
  and not for representations.[133X
  
  [33X[0;0YWhen  [13Xcreating[113X new rational functions however they must be created in one of
  the three representations. In most cases this will be the representation for
  which the [21Xconceptual[121X representation in which the calculation was done is the
  defining attribute.[133X
  
  [33X[0;0YIterated  operations  (like  forming the product over a list) therefore will
  tend  to  stay  in  the  most suitable representation and the calculation of
  another  conceptual  representation (which may be comparatively expensive in
  certain circumstances) is not necessary.[133X
  
  
  [1X66.21 [33X[0;0YThe Defining Attributes of Rational Functions[133X[101X
  
  [33X[0;0YIn  general,  rational  functions  are given in terms of monomials. They are
  represented by lists, using numbers (see [14X66.1[114X) for the indeterminates.[133X
  
  [33X[0;0YA monomial is a product of powers of indeterminates. A monomial is stored as
  a  list  (we  call  this  the  [13Xexpanded  form[113X  of  the monomial) of the form
  [10X[[3Xinum[103X[10X,[3Xexp[103X[10X,[3Xinum[103X[10X,[3Xexp[103X[10X,...][110X  where  each  [3Xinum[103X is the number of an indeterminate
  and [3Xexp[103X the corresponding exponent. The list must be sorted according to the
  numbers of the indeterminates. Thus for example, if [22Xx[122X, [22Xy[122X and [22Xz[122X are the first
  three indeterminates, the expanded form of the monomial [22Xx^5 z^8 = z^8 x^5[122X is
  [10X[  1,  5,  3, 8 ][110X. The representation of a polynomials is a list of the form
  [10X[[3Xmon[103X[10X,[3Xcoeff[103X[10X,[3Xmon[103X[10X,[3Xcoeff[103X[10X,...][110X  where [3Xmon[103X is a monomial in expanded form (that is
  given  as  list)  and  [3Xcoeff[103X  its  coefficient. The monomials must be sorted
  according to the total degree/lexicographic order (This is the same as given
  by  the  [21Xgrlex[121X  monomial  ordering, see [2XMonomialGrlexOrdering[102X ([14X66.17-8[114X)). We
  call  this  the  [13Xexternal  representation[113X  of  a polynomial. (The reason for
  ordering  is  that  addition  of polynomials becomes linear in the number of
  monomials instead of quadratic; the reason for the particular ordering chose
  is  that  it  is  compatible  with  multiplication and thus gives acceptable
  performance for quotient calculations.)[133X
  
  [33X[0;0YThe  attributes  that  give  a  representation of a a rational function as a
  Laurent   polynomial   are   [2XCoefficientsOfLaurentPolynomial[102X  ([14X66.13-2[114X)  and
  [2XIndeterminateNumberOfUnivariateRationalFunction[102X ([14X66.1-2[114X).[133X
  
  [33X[0;0YAlgorithms  should  use only the attributes [2XExtRepNumeratorRatFun[102X ([14X66.21-2[114X),
  [2XExtRepDenominatorRatFun[102X    ([14X66.21-3[114X),    [2XExtRepPolynomialRatFun[102X   ([14X66.21-6[114X),
  [2XCoefficientsOfLaurentPolynomial[102X ([14X66.13-2[114X) and –if the univariate function is
  not  constant–  [2XIndeterminateNumberOfUnivariateRationalFunction[102X  ([14X66.1-2[114X) as
  the  low-level interface to work with a polynomial. They should not refer to
  the actual representation used.[133X
  
  [1X66.21-1 IsRationalFunctionDefaultRep[101X
  
  [29X[2XIsRationalFunctionDefaultRep[102X( [3Xobj[103X ) [32X Representation
  
  [33X[0;0Yis  the default representation of rational functions. A rational function in
  this  representation  is  defined  by  the  attributes [2XExtRepNumeratorRatFun[102X
  ([14X66.21-2[114X)  and  [2XExtRepDenominatorRatFun[102X  ([14X66.21-3[114X),  the values of which are
  external representations of polynomials.[133X
  
  [1X66.21-2 ExtRepNumeratorRatFun[101X
  
  [29X[2XExtRepNumeratorRatFun[102X( [3Xratfun[103X ) [32X attribute
  
  [33X[0;0Yreturns  the  external  representation  of  the  numerator polynomial of the
  rational function [3Xratfun[103X. Numerator and denominator are not guaranteed to be
  cancelled against each other.[133X
  
  [1X66.21-3 ExtRepDenominatorRatFun[101X
  
  [29X[2XExtRepDenominatorRatFun[102X( [3Xratfun[103X ) [32X attribute
  
  [33X[0;0Yreturns  the  external  representation  of the denominator polynomial of the
  rational function [3Xratfun[103X. Numerator and denominator are not guaranteed to be
  cancelled against each other.[133X
  
  [1X66.21-4 ZeroCoefficientRatFun[101X
  
  [29X[2XZeroCoefficientRatFun[102X( [3Xratfun[103X ) [32X operation
  
  [33X[0;0Yreturns  the zero of the coefficient ring. This might be needed to represent
  the  zero  polynomial for which the external representation of the numerator
  is the empty list.[133X
  
  [1X66.21-5 IsPolynomialDefaultRep[101X
  
  [29X[2XIsPolynomialDefaultRep[102X( [3Xobj[103X ) [32X Representation
  
  [33X[0;0Yis   the  default  representation  of  polynomials.  A  polynomial  in  this
  representation  is  defined  by  the  components  and  [2XExtRepNumeratorRatFun[102X
  ([14X66.21-2[114X)    where   [2XExtRepNumeratorRatFun[102X   ([14X66.21-2[114X)   is   the   external
  representation of the polynomial.[133X
  
  [1X66.21-6 ExtRepPolynomialRatFun[101X
  
  [29X[2XExtRepPolynomialRatFun[102X( [3Xpolynomial[103X ) [32X attribute
  
  [33X[0;0Yreturns  the  external  representation  of  a  polynomial. The difference to
  [2XExtRepNumeratorRatFun[102X  ([14X66.21-2[114X) is that rational functions might know to be
  a  polynomial  but  can still have a non-vanishing denominator. In this case
  [2XExtRepPolynomialRatFun[102X has to call a quotient routine.[133X
  
  [1X66.21-7 IsLaurentPolynomialDefaultRep[101X
  
  [29X[2XIsLaurentPolynomialDefaultRep[102X( [3Xobj[103X ) [32X Representation
  
  [33X[0;0YThis   representation   is  used  for  Laurent  polynomials  and  univariate
  polynomials.   It   represents  a  Laurent  polynomial  via  the  attributes
  [2XCoefficientsOfLaurentPolynomial[102X                 ([14X66.13-2[114X)                and
  [2XIndeterminateNumberOfLaurentPolynomial[102X ([14X66.13-3[114X).[133X
  
  
  [1X66.22 [33X[0;0YCreation of Rational Functions[133X[101X
  
  [33X[0;0YThe operations [2XLaurentPolynomialByCoefficients[102X ([14X66.13-1[114X), [2XPolynomialByExtRep[102X
  ([14X66.22-2[114X)  and  [2XRationalFunctionByExtRep[102X  ([14X66.22-1[114X)  are  used  to construct
  objects in the three basic representations for rational functions.[133X
  
  [1X66.22-1 RationalFunctionByExtRep[101X
  
  [29X[2XRationalFunctionByExtRep[102X( [3Xrfam[103X, [3Xnum[103X, [3Xden[103X ) [32X function
  [29X[2XRationalFunctionByExtRepNC[102X( [3Xrfam[103X, [3Xnum[103X, [3Xden[103X ) [32X function
  
  [33X[0;0Yconstructs     a     rational     function     (in     the    representation
  [2XIsRationalFunctionDefaultRep[102X  ([14X66.21-1[114X))  in  the  rational  function family
  [3Xrfam[103X,  the rational function itself is given by the external representations
  [3Xnum[103X and [3Xden[103X for numerator and denominator. No cancellation takes place.[133X
  
  [33X[0;0YThe  variant  [2XRationalFunctionByExtRepNC[102X  does  not  perform any test of the
  arguments and thus potentially can create illegal objects. It only should be
  used if speed is required and the arguments are known to be in correct form.[133X
  
  [1X66.22-2 PolynomialByExtRep[101X
  
  [29X[2XPolynomialByExtRep[102X( [3Xrfam[103X, [3Xextrep[103X ) [32X function
  [29X[2XPolynomialByExtRepNC[102X( [3Xrfam[103X, [3Xextrep[103X ) [32X function
  
  [33X[0;0Yconstructs   a  polynomial  (in  the  representation  [2XIsPolynomialDefaultRep[102X
  ([14X66.21-5[114X))  in  the  rational function family [3Xrfam[103X, the polynomial itself is
  given by the external representation [3Xextrep[103X.[133X
  
  [33X[0;0YThe  variant [2XPolynomialByExtRepNC[102X does not perform any test of the arguments
  and  thus  potentially can create invalid objects. It only should be used if
  speed is required and the arguments are known to be in correct form.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfam:=RationalFunctionsFamily(FamilyObj(1));;[127X[104X
    [4X[25Xgap>[125X [27Xp:=PolynomialByExtRep(fam,[[1,2],1,[2,1,15,7],3]);[127X[104X
    [4X[28X3*y*x_15^7+x^2[128X[104X
    [4X[25Xgap>[125X [27Xq:=p/(p+1);[127X[104X
    [4X[28X(3*y*x_15^7+x^2)/(3*y*x_15^7+x^2+1)[128X[104X
    [4X[25Xgap>[125X [27XExtRepNumeratorRatFun(q);[127X[104X
    [4X[28X[ [ 1, 2 ], 1, [ 2, 1, 15, 7 ], 3 ][128X[104X
    [4X[25Xgap>[125X [27XExtRepDenominatorRatFun(q);[127X[104X
    [4X[28X[ [  ], 1, [ 1, 2 ], 1, [ 2, 1, 15, 7 ], 3 ][128X[104X
  [4X[32X[104X
  
  [1X66.22-3 LaurentPolynomialByExtRep[101X
  
  [29X[2XLaurentPolynomialByExtRep[102X( [3Xfam[103X, [3Xcofs[103X, [3Xval[103X, [3Xind[103X ) [32X function
  [29X[2XLaurentPolynomialByExtRepNC[102X( [3Xfam[103X, [3Xcofs[103X, [3Xval[103X, [3Xind[103X ) [32X function
  
  [33X[0;0Ycreates  a  Laurent polynomial in the family [3Xfam[103X with [[3Xcofs[103X,[3Xval[103X] as value of
  [2XCoefficientsOfLaurentPolynomial[102X   ([14X66.13-2[114X).   No  coefficient  shifting  is
  performed.  This is the lowest level function to create a Laurent polynomial
  but  will  rely  on  the  coefficients  being  shifted properly and will not
  perform   any   tests.   Unless  this  is  guaranteed  for  the  parameters,
  [2XLaurentPolynomialByCoefficients[102X ([14X66.13-1[114X) should be used.[133X
  
  
  [1X66.23 [33X[0;0YArithmetic for External Representations of Polynomials[133X[101X
  
  [33X[0;0YThe  following  operations are used internally to perform the arithmetic for
  polynomials in their [21Xexternal[121X representation (see [14X66.21[114X) as lists.[133X
  
  [33X[0;0YFunctions  to  perform  arithmetic  with  the  coefficient  lists of Laurent
  polynomials are described in Section [14X23.4[114X.[133X
  
  [1X66.23-1 ZippedSum[101X
  
  [29X[2XZippedSum[102X( [3Xz1[103X, [3Xz2[103X, [3Xczero[103X, [3Xfuncs[103X ) [32X operation
  
  [33X[0;0Ycomputes  the  sum of two external representations of polynomials [3Xz1[103X and [3Xz2[103X.
  [3Xczero[103X  is the appropriate coefficient zero and [3Xfuncs[103X a list [ [3Xmonomial_less[103X,
  [3Xcoefficient_sum[103X  ]  containing  a  monomial  comparison  and  a  coefficient
  addition function. This list can be found in the component [3Xfam[103X[10X!.zippedSum[110X of
  the rational functions family.[133X
  
  [33X[0;0YNote  that  [3Xcoefficient_sum[103X  must  be  a  proper  [21Xsummation[121X  function, not a
  function computing differences.[133X
  
  [1X66.23-2 ZippedProduct[101X
  
  [29X[2XZippedProduct[102X( [3Xz1[103X, [3Xz2[103X, [3Xczero[103X, [3Xfuncs[103X ) [32X operation
  
  [33X[0;0Ycomputes  the  product of two external representations of polynomials [3Xz1[103X and
  [3Xz2[103X.   [3Xczero[103X  is  the  appropriate  coefficient  zero  and  [3Xfuncs[103X  a  list  [
  [3Xmonomial_prod[103X,  [3Xmonomial_less[103X, [3Xcoefficient_sum[103X, [3Xcoefficient_prod[103X] containing
  functions  to  multiply  and  compare  monomials,  to  add  and  to multiply
  coefficients.  This list can be found in the component [10X[3Xfam[103X[10X!.zippedProduct[110X of
  the rational functions family.[133X
  
  [1X66.23-3 QuotientPolynomialsExtRep[101X
  
  [29X[2XQuotientPolynomialsExtRep[102X( [3Xfam[103X, [3Xa[103X, [3Xb[103X ) [32X function
  
  [33X[0;0YLet  [3Xa[103X and [3Xb[103X the external representations of two polynomials in the rational
  functions  family [3Xfam[103X. This function computes the external representation of
  the  quotient  of  both  polynomials,  it  returns  [9Xfail[109X  if  the polynomial
  described by [3Xb[103X does not divide the polynomial described by [3Xa[103X.[133X
  
  
  [1X66.24 [33X[0;0YCancellation Tests for Rational Functions[133X[101X
  
  [33X[0;0YThe  operation  [2XGcd[102X  ([14X56.7-1[114X)  can be used to test for common factors of two
  polynomials.  This  however  would  be  too  expensive  to  be  done  in the
  arithmetic, thus uses the following operations internally to try to keep the
  denominators as small as possible[133X
  
  [1X66.24-1 RationalFunctionByExtRepWithCancellation[101X
  
  [29X[2XRationalFunctionByExtRepWithCancellation[102X( [3Xrfam[103X, [3Xnum[103X, [3Xden[103X ) [32X function
  
  [33X[0;0Yconstructs  a  rational  function as [2XRationalFunctionByExtRep[102X ([14X66.22-1[114X) does
  but tries to cancel out common factors of numerator and denominator, calling
  [2XTryGcdCancelExtRepPolynomials[102X ([14X66.24-2[114X).[133X
  
  [1X66.24-2 TryGcdCancelExtRepPolynomials[101X
  
  [29X[2XTryGcdCancelExtRepPolynomials[102X( [3Xfam[103X, [3Xa[103X, [3Xb[103X ) [32X function
  
  [33X[0;0YLet  [3Xa[103X  and  [3Xb[103X  be  the  external  representations  of two polynomials. This
  function   tries   to   cancel  common  factors  between  the  corresponding
  polynomials  and  returns  a  list [22X[ a', b' ][122X of external representations of
  cancelled  polynomials.  As there is no proper multivariate GCD cancellation
  is not guaranteed to be optimal.[133X
  
  [1X66.24-3 HeuristicCancelPolynomials[101X
  
  [29X[2XHeuristicCancelPolynomials[102X( [3Xfam[103X, [3Xext1[103X, [3Xext2[103X ) [32X operation
  
  [33X[0;0Yis  called  by [2XTryGcdCancelExtRepPolynomials[102X ([14X66.24-2[114X) to perform the actual
  work.   It   will   return  either  [9Xfail[109X  or  a  new  list  of  of  external
  representations  of cancelled polynomials. The cancellation performed is not
  necessarily optimal.[133X
  
