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66 Polynomials and Rational Functions
 66.1 Indeterminates
 66.2 Operations for Rational Functions
 66.3 Comparison of Rational Functions
 66.4 Properties and Attributes of Rational Functions
 66.5 Univariate Polynomials
 66.6 Polynomials as Univariate Polynomials in one Indeterminate
 66.7 Multivariate Polynomials
 66.8 Minimal Polynomials
 66.9 Cyclotomic Polynomials
 66.10 Polynomial Factorization
 66.11 Polynomials over the Rationals
 66.12 Factorization of Polynomials over the Rationals
 66.13 Laurent Polynomials
 66.14 Univariate Rational Functions
 66.15 Polynomial Rings and Function Fields
 66.16 Univariate Polynomial Rings
 66.17 Monomial Orderings
 66.18 Groebner Bases
 66.19 Rational Function Families
 66.20 The Representations of Rational Functions
 66.21 The Defining Attributes of Rational Functions
 66.22 Creation of Rational Functions
 66.23 Arithmetic for External Representations of Polynomials
 66.24 Cancellation Tests for Rational Functions

66 Polynomials and Rational Functions

Let \(R\) be a commutative ring-with-one. We call a free associative algebra \(A\) over \(R\) a polynomial ring over \(R\). The free generators of \(A\) are called indeterminates (to avoid naming conflicts with the word variables which will be used to denote GAP variables only) , they are usually denoted by \(x_1, x_2, \ldots\). The number of indeterminates is called the rank of \(A\). The elements of \(A\) are called polynomials. Products of indeterminates are called monomials, every polynomial can be expressed as a finite sum of products of monomials with ring elements in a form like \(r_{{1,0}} x_1 + r_{{1,1}} x_1 x_2 + r_{{0,1}} x_2 + \cdots\) with \(r_{{i,j}} \in R\).

A polynomial ring of rank 1 is called an univariate polynomial ring, its elements are univariate polynomials.

Polynomial rings of smaller rank naturally embed in rings of higher rank; if \(S\) is a subring of \(R\) then a polynomial ring over \(S\) naturally embeds in a polynomial ring over \(R\) of the same rank. Note however that GAP does not consider \(R\) as a subset of a polynomial ring over \(R\); for example the zero of \(R\) (\(0\)) and the zero of the polynomial ring (\(0x^0\)) are different objects.

Internally, 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 66.1 for details.

If \(R\) is an integral domain, the polynomial ring \(A\) over \(R\) is an integral domain as well and one can therefore form its quotient field \(Q\). This field is called a field of rational functions. Again \(A\) embeds naturally into \(Q\) and GAP will perform this embedding implicitly. (In fact it implements the ring of rational functions over \(R\).) To avoid problems with leading coefficients, however, \(R\) must be a unique factorization domain.

66.1 Indeterminates

Internally, indeterminates are created for a family of objects (for example all elements of finite fields in characteristic \(3\) are in one family). Thus a variable "x" over the rationals is also an "x" over the integers, while an "x" over GF(3) is different.

Within one family, every indeterminate has a number nr and as long as no other names have been assigned, this indeterminate will be displayed as "x_nr". Indeterminate numbers can be arbitrary nonnegative integers.

It 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 "x_nr".

(Because of this printing convention, the name x_nr is interpreted specially to always denote the variable with internal number nr.)

The indeterminate names have not necessarily any relations to variable names: this means that an indeterminate whose name is, say, "x" cannot be accessed using the variable x, unless x was defined to be that indeterminate.

When asking for indeterminates with certain names, GAP 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 IndeterminateNumberOfLaurentPolynomial(indet) will return the number of the indeterminate indet.

When asked to create an indeterminate with a name that exists already for the family, GAP will by default return this existing indeterminate. If you explicitly want a new indeterminate, distinct from the already existing one with the same name, you can add the new option to the function call. (This is in most cases not a good idea.)

gap> R:=PolynomialRing(GF(3),["x","y","z"]);
GF(3)[x,y,z]
gap> List(IndeterminatesOfPolynomialRing(R),
>   IndeterminateNumberOfLaurentPolynomial);
[ 1, 2, 3 ]
gap> R:=PolynomialRing(GF(3),["z"]);
GF(3)[z]
gap> List(IndeterminatesOfPolynomialRing(R),
>   IndeterminateNumberOfLaurentPolynomial);
[ 3 ]
gap> R:=PolynomialRing(GF(3),["x","y","z"]:new);
GF(3)[x,y,z]
gap> List(IndeterminatesOfPolynomialRing(R),
>   IndeterminateNumberOfLaurentPolynomial);
[ 4, 5, 6 ]
gap> R:=PolynomialRing(GF(3),["z"]);
GF(3)[z]
gap> List(IndeterminatesOfPolynomialRing(R),
>   IndeterminateNumberOfLaurentPolynomial);
[ 3 ]

66.1-1 Indeterminate
‣ Indeterminate( R[, nr] )( operation )
‣ Indeterminate( R[, name][, avoid] )( operation )
‣ Indeterminate( fam, nr )( operation )
‣ X( R[, nr] )( operation )
‣ X( R[, name][, avoid] )( operation )
‣ X( fam, nr )( operation )

returns the indeterminate number nr over the ring R. If nr is not given it defaults to 1. If the number is not specified a list avoid of indeterminates may be given. The function will return an indeterminate that is guaranteed to be different from all the indeterminates in the list avoid. The third usage returns an indeterminate called name (also avoiding the indeterminates in avoid if given).

X is simply a synonym for Indeterminate. However, we do not recommend to use this synonym which is supported only for the backwards compatibility.

gap> x:=Indeterminate(GF(3),"x");
x
gap> y:=X(GF(3),"y");z:=X(GF(3),"X");
y
X
gap> X(GF(3),2);
y
gap> X(GF(3),"x_3");
X
gap> X(GF(3),[y,z]);
x

66.1-2 IndeterminateNumberOfUnivariateRationalFunction
‣ IndeterminateNumberOfUnivariateRationalFunction( rfun )( attribute )

returns the number of the indeterminate in which the univariate rational function rfun is expressed. (This also provides a way to obtain the number of a given indeterminate.)

A constant rational function might not possess an indeterminate number. In this case IndeterminateNumberOfUnivariateRationalFunction 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 CIUnivPols (66.1-5) and IsLaurentPolynomialDefaultRep (66.21-7)).

66.1-3 IndeterminateOfUnivariateRationalFunction
‣ IndeterminateOfUnivariateRationalFunction( rfun )( attribute )

returns the indeterminate in which the univariate rational function rfun is expressed. (cf. IndeterminateNumberOfUnivariateRationalFunction (66.1-2).)

gap> IndeterminateNumberOfUnivariateRationalFunction(z);
3
gap> IndeterminateOfUnivariateRationalFunction(z^5+z);
X

66.1-4 IndeterminateName
‣ IndeterminateName( fam, nr )( operation )
‣ HasIndeterminateName( fam, nr )( operation )
‣ SetIndeterminateName( fam, nr, name )( operation )

SetIndeterminateName assigns the name name to indeterminate nr in the rational functions family fam. It issues an error if the indeterminate was already named.

IndeterminateName returns the name of the nr-th indeterminate (and returns fail if no name has been assigned).

HasIndeterminateName tests whether indeterminate nr has already been assigned a name.

gap> IndeterminateName(FamilyObj(x),2);
"y"
gap> HasIndeterminateName(FamilyObj(x),4);
false
gap> SetIndeterminateName(FamilyObj(x),10,"bla");
gap> Indeterminate(GF(3),10);
bla

As a convenience there is a special method installed for SetName that will assign a name to an indeterminate.

gap> a:=Indeterminate(GF(3),5);
x_5
gap> SetName(a,"ah");
gap> a^5+a;
ah^5+ah

66.1-5 CIUnivPols
‣ CIUnivPols( upol1, upol2 )( function )

This function (whose name stands for "common indeterminate of univariate polynomials") takes two univariate polynomials as arguments. If both polynomials are given in the same indeterminate number indnum (in this case they are "compatible" as univariate polynomials) it returns indnum. In all other cases it returns fail. CIUnivPols also accepts if either polynomial is constant but formally expressed in another indeterminate, in this situation the indeterminate of the other polynomial is selected.

66.2 Operations for Rational Functions

The rational functions form a field, therefore all arithmetic operations are applicable to rational functions.

f + g

f - g

f * g

f / g

gap> x:=Indeterminate(Rationals,1);;y:=Indeterminate(Rationals,2);;
gap> f:=3+x*y+x^5;;g:=5+x^2*y+x*y^2;;
gap> a:=g/f;
(x_1^2*x_2+x_1*x_2^2+5)/(x_1^5+x_1*x_2+3)

Note that the quotient f/g of two polynomials might be represented as a rational function again. If g is known to divide f the call Quotient(f,g) (see Quotient (56.1-9)) should be used instead.

f mod g

For two Laurent polynomials f and g, f mod g is the Euclidean remainder (see EuclideanRemainder (56.6-4)) of f modulo g.

As 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 Gcd (56.7-1) on the NumeratorOfRationalFunction (66.4-2) and DenominatorOfRationalFunction (66.4-3) values of a given rational function.

All 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.

66.3 Comparison of Rational Functions

f = g

Two rational functions f and g are equal if the product Numerator(f) * Denominator(g) equals Numerator(g) * Denominator(f).

gap> x:=Indeterminate(Rationals,"x");;y:=Indeterminate(Rationals,"y");;
gap> f:=3+x*y+x^5;;g:=5+x^2*y+x*y^2;;
gap> a:=g/f;
(x^2*y+x*y^2+5)/(x^5+x*y+3)
gap> b:=(g*f)/(f^2);
(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\
*x^6*y+6*x^5+x^2*y^2+6*x*y+9)
gap> a=b;
true

f < g

The ordering of rational functions is defined in several steps. Monomials (products of indeterminates) are sorted first by degree, then lexicographically (with \(x_1>x_2\)) (see MonomialGrlexOrdering (66.17-8)). Products of monomials with ring elements ("terms") are compared first by their monomials and then by their coefficients.

gap> x>y;
true
gap> x^2*y<x*y^2;
false
gap> x*y<x^2*y;
true
gap> x^2*y < 5* y*x^2;
true

Polynomials are compared by comparing the largest terms in turn until they differ.

gap> x+y<y;
false
gap> x<x+1;
true

Rational functions are compared by comparing the polynomial Numerator(f) * Denominator(g) with the polynomial Numerator(g) * Denominator(f). (As the ordering of monomials used by GAP is invariant under multiplication this is independent of common factors in numerator and denominator.)

gap> f/g<g/f;
false
gap> f/g<(g*g)/(f*g);
false

For univariate polynomials this reduces to an ordering first by total degree and then lexicographically on the coefficients.

66.4 Properties and Attributes of Rational Functions

All these tests are applicable to every 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.

For reasons of performance within algorithms it can be useful to use other attributes, which give a slightly more technical representation. See section 66.20 for details.

66.4-1 IsPolynomialFunction
‣ IsPolynomialFunction( obj )( category )
‣ IsRationalFunction( obj )( category )

A 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 NumeratorOfRationalFunction (66.4-2)) and its denominator (see DenominatorOfRationalFunction (66.4-3))

A polynomial function is an element of a polynomial ring (not necessarily an UFD), or a rational function.

GAP considers IsRationalFunction as a subcategory of IsPolynomialFunction.

66.4-2 NumeratorOfRationalFunction
‣ NumeratorOfRationalFunction( ratfun )( attribute )

returns the numerator of the rational function ratfun.

As no proper multivariate gcd has been implemented yet, numerators and denominators are not guaranteed to be reduced!

66.4-3 DenominatorOfRationalFunction
‣ DenominatorOfRationalFunction( ratfun )( attribute )

returns the denominator of the rational function ratfun.

As no proper multivariate gcd has been implemented yet, numerators and denominators are not guaranteed to be reduced!

gap> x:=Indeterminate(Rationals,1);;y:=Indeterminate(Rationals,2);;
gap> DenominatorOfRationalFunction((x*y+x^2)/y);
y
gap> NumeratorOfRationalFunction((x*y+x^2)/y);
x^2+x*y

66.4-4 IsPolynomial
‣ IsPolynomial( ratfun )( property )

A polynomial is a rational function whose denominator is one. (If the coefficients family forms a field this is equivalent to the denominator being constant.)

If 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.

gap> IsPolynomial((x*y+x^2*y^3)/y);
true
gap> IsPolynomial((x*y+x^2)/y);
false

66.4-5 AsPolynomial
‣ AsPolynomial( poly )( attribute )

If poly is a rational function that is a polynomial this attribute returns an equal rational function \(p\) such that \(p\) is equal to its numerator and the denominator of \(p\) is one.

gap> AsPolynomial((x*y+x^2*y^3)/y);
x^2*y^2+x

66.4-6 IsUnivariateRationalFunction
‣ IsUnivariateRationalFunction( ratfun )( property )

A rational function is univariate if its numerator and its denominator are both polynomials in the same one indeterminate. The attribute IndeterminateNumberOfUnivariateRationalFunction (66.1-2) can be used to obtain the number of this common indeterminate.

66.4-7 CoefficientsOfUnivariateRationalFunction
‣ CoefficientsOfUnivariateRationalFunction( rfun )( attribute )

if rfun is a univariate rational function, this attribute returns a list [ ncof, dcof, val ] where ncof and dcof are coefficient lists of univariate polynomials n and d and a valuation val such that \(\textit{rfun} = x^{\textit{val}} \cdot \textit{n} / \textit{d}\) where \(x\) is the variable with the number given by IndeterminateNumberOfUnivariateRationalFunction (66.1-2). Numerator and denominator are guaranteed to be cancelled.

66.4-8 IsUnivariatePolynomial
‣ IsUnivariatePolynomial( ratfun )( property )

A univariate polynomial is a polynomial in only one indeterminate.

66.4-9 CoefficientsOfUnivariatePolynomial
‣ CoefficientsOfUnivariatePolynomial( pol )( attribute )

CoefficientsOfUnivariatePolynomial returns the coefficient list of the polynomial pol, sorted in ascending order. (It returns the empty list if pol is 0.)

66.4-10 IsLaurentPolynomial
‣ IsLaurentPolynomial( ratfun )( property )

A Laurent polynomial is a univariate rational function whose denominator is a monomial. Therefore every univariate polynomial is a Laurent polynomial.

The attribute CoefficientsOfLaurentPolynomial (66.13-2) gives a compact representation as Laurent polynomial.

66.4-11 IsConstantRationalFunction
‣ IsConstantRationalFunction( ratfun )( property )

A constant rational function is a function whose numerator and denominator are polynomials of degree 0.

66.4-12 IsPrimitivePolynomial
‣ IsPrimitivePolynomial( F, pol )( operation )

For a univariate polynomial pol of degree \(d\) in the indeterminate \(X\), with coefficients in a finite field F with \(q\) elements, say, IsPrimitivePolynomial returns true if

  1. pol divides \(X^{{q^d-1}} - 1\), and

  2. for each prime divisor \(p\) of \(q^d - 1\), pol does not divide \(X^{{(q^d-1)/p}} - 1\),

and false otherwise.

66.4-13 SplittingField
‣ SplittingField( f )( attribute )

returns the smallest field which contains the coefficients of f and the roots of f.

66.5 Univariate Polynomials

Some of the operations are actually defined on the larger domain of Laurent polynomials (see 66.13). For this section you can simply ignore the word "Laurent" if it occurs in a description.

66.5-1 UnivariatePolynomial
‣ UnivariatePolynomial( ring, cofs[, ind] )( operation )

constructs an univariate polynomial over the ring ring in the indeterminate ind with the coefficients given by coefs.

66.5-2 UnivariatePolynomialByCoefficients
‣ UnivariatePolynomialByCoefficients( fam, cofs, ind )( operation )

constructs an univariate polynomial over the coefficients family fam and in the indeterminate ind with the coefficients given by coefs. This function should be used in algorithms to create polynomials as it avoids overhead associated with UnivariatePolynomial (66.5-1).

66.5-3 DegreeOfLaurentPolynomial
‣ DegreeOfLaurentPolynomial( pol )( attribute )

The degree of a univariate (Laurent) polynomial pol is the largest exponent \(n\) of a monomial \(x^n\) of pol. The degree of a zero polynomial is defined to be -infinity.

gap> p:=UnivariatePolynomial(Rationals,[1,2,3,4],1);
4*x^3+3*x^2+2*x+1
gap> UnivariatePolynomialByCoefficients(FamilyObj(1),[9,2,3,4],73);
4*x_73^3+3*x_73^2+2*x_73+9
gap> CoefficientsOfUnivariatePolynomial(p);
[ 1, 2, 3, 4 ]
gap> DegreeOfLaurentPolynomial(p);
3
gap> DegreeOfLaurentPolynomial(Zero(p));
-infinity
gap> IndeterminateNumberOfLaurentPolynomial(p);
1
gap> IndeterminateOfLaurentPolynomial(p);
x

66.5-4 RootsOfPolynomial
‣ RootsOfPolynomial( [R, ]p )( function )

For a univariate polynomial p, this function returns all roots of p over the ring R. If the ring is not specified, it defaults to the ring specified by the coefficients of p via DefaultRing (56.1-3)).

gap> x:=X(Rationals,"x");;p:=x^4-1;    
x^4-1
gap> RootsOfPolynomial(p);
[ 1, -1 ]
gap> RootsOfPolynomial(CF(4),p);
[ 1, -1, E(4), -E(4) ]

66.5-5 RootsOfUPol
‣ RootsOfUPol( [field, ]upol )( function )

This function returns a list of all roots of the univariate polynomial upol in its default domain. If the optional argument field is a field then the roots in this field are computed. If field is the string "split" then the splitting field of the polynomial is taken.

gap> RootsOfUPol(50-45*x-6*x^2+x^3);
[ 10, 1, -5 ]

66.5-6 QuotRemLaurpols
‣ QuotRemLaurpols( left, right, mode )( function )

This internal function for euclidean division of polynomials takes two polynomials left and right and computes their quotient. No test is performed whether the arguments indeed are polynomials. Depending on the integer variable mode, which may take values in a range from 1 to 4, it returns respectively:

  1. the quotient (there might be some remainder),

  2. the remainder,

  3. a list [q,r] of quotient and remainder,

  4. the quotient if there is no remainder and fail otherwise.

66.5-7 UnivariatenessTestRationalFunction
‣ UnivariatenessTestRationalFunction( f )( function )

takes a rational function f and tests whether it is univariate rational function (or even a Laurent polynomial). It returns a list [isunivariate, indet, islaurent, cofs].

If f is a univariate rational function then isunivariate is true and indet is the number of the appropriate indeterminate.

Furthermore, if f is a Laurent polynomial, then islaurent is also true. In this case the fourth entry, cofs, is the value of the attribute CoefficientsOfLaurentPolynomial (66.13-2) for f.

If isunivariate is true but islaurent is false, then cofs is the value of the attribute CoefficientsOfUnivariateRationalFunction (66.4-7) for f.

Otherwise, each entry of the returned list is equal to fail. As there is no proper multivariate gcd, this may also happen for the rational function which may be reduced to univariate (see example).

gap> UnivariatenessTestRationalFunction( 50-45*x-6*x^2+x^3 );
[ true, 1, true, [ [ 50, -45, -6, 1 ], 0 ] ]
gap> UnivariatenessTestRationalFunction( (-6*y^2+y^3) / (y+1) );
[ true, 2, false, [ [ -6, 1 ], [ 1, 1 ], 2 ] ]
gap> UnivariatenessTestRationalFunction( (-6*y^2+y^3) / (x+1));
[ fail, fail, fail, fail ]
gap> UnivariatenessTestRationalFunction( ((y+2)*(x+1)) / ((y-1)*(x+1)) );
[ fail, fail, fail, fail ]

66.5-8 InfoPoly
‣ InfoPoly( info class )

is the info class for univariate polynomials.

We 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 Value (66.7-1), Discriminant (66.6-6), Derivative (66.6-5), LeadingCoefficient (66.6-3) and LeadingMonomial (66.6-4):

gap> p:=UnivariatePolynomial(Rationals,[1,2,3,4],1);
4*x^3+3*x^2+2*x+1
gap> Value(p,Z(5));
Z(5)^2
gap> LeadingCoefficient(p);
4
gap> Derivative(p);
12*x^2+6*x+2

66.6 Polynomials as Univariate Polynomials in one Indeterminate

66.6-1 DegreeIndeterminate
‣ DegreeIndeterminate( pol, ind )( operation )

returns the degree of the polynomial pol in the indeterminate (or indeterminate number) ind.

gap> f:=x^5+3*x*y+9*y^7+4*y^5*x+3*y+2;
9*y^7+4*x*y^5+x^5+3*x*y+3*y+2
gap> DegreeIndeterminate(f,1);
5
gap> DegreeIndeterminate(f,y);
7

66.6-2 PolynomialCoefficientsOfPolynomial
‣ PolynomialCoefficientsOfPolynomial( pol, ind )( operation )

PolynomialCoefficientsOfPolynomial returns the coefficient list (whose entries are polynomials not involving the indeterminate ind) describing the polynomial pol viewed as a polynomial in ind. Instead of the indeterminate, ind can also be an indeterminate number.

gap> PolynomialCoefficientsOfPolynomial(f,2);
[ x^5+2, 3*x+3, 0, 0, 0, 4*x, 0, 9 ]

66.6-3 LeadingCoefficient
‣ LeadingCoefficient( pol )( operation )

returns the leading coefficient (that is the coefficient of the leading monomial, see LeadingMonomial (66.6-4)) of the polynomial pol.

66.6-4 LeadingMonomial
‣ LeadingMonomial( pol )( operation )

returns the leading monomial (with respect to the ordering given by MonomialExtGrlexLess (66.17-14)) of the polynomial pol as a list containing indeterminate numbers and exponents.

gap> LeadingCoefficient(f,1);
1
gap> LeadingCoefficient(f,2);
9
gap> LeadingMonomial(f);
[ 2, 7 ]
gap> LeadingCoefficient(f);
9

66.6-5 Derivative
‣ Derivative( ratfun[, ind] )( attribute )

If ratfun is a univariate rational function then Derivative returns the derivative of ufun by its indeterminate. For a rational function ratfun, the derivative by the indeterminate ind is returned, regarding ratfun as univariate in ind. Instead of the desired indeterminate, also the number of this indeterminate can be given as ind.

gap> Derivative(f,2);
63*y^6+20*x*y^4+3*x+3

66.6-6 Discriminant
‣ Discriminant( pol[, ind] )( operation )

If pol is a univariate polynomial then Discriminant returns the discriminant of pol by its indeterminate. The two-argument form returns the discriminant of a polynomial pol by the indeterminate number ind, regarding pol as univariate in this indeterminate. Instead of the indeterminate number, the indeterminate itself can also be given as ind.

gap> Discriminant(f,1);
20503125*y^28+262144*y^25+27337500*y^22+19208040*y^21+1474560*y^17+136\
68750*y^16+18225000*y^15+6075000*y^14+1105920*y^13+3037500*y^10+648972\
0*y^9+4050000*y^8+900000*y^7+62208*y^5+253125*y^4+675000*y^3+675000*y^\
2+300000*y+50000
gap> Discriminant(f,1) = Discriminant(f,x);
true

66.6-7 Resultant
‣ Resultant( pol1, pol2, ind )( operation )

computes the resultant of the polynomials pol1 and pol2 with respect to the indeterminate ind, or indeterminate number ind. The resultant considers pol1 and pol2 as univariate in ind and returns an element of the corresponding base ring (which might be a polynomial ring).

gap> Resultant(x^4+y,y^4+x,1);
y^16+y
gap> Resultant(x^4+y,y^4+x,2);
x^16+x

66.7 Multivariate Polynomials

66.7-1 Value
‣ Value( ratfun, indets, vals[, one] )( operation )
‣ Value( upol, value[, one] )( operation )

The first variant takes a rational function ratfun and specializes the indeterminates given in indets to the values given in vals, replacing the \(i\)-th entry in indets by the \(i\)-th entry in vals. If this specialization results in a constant polynomial, an element of the coefficient ring is returned. If the specialization would specialize the denominator of ratfun to zero, an error is raised.

A variation is the evaluation at elements of another ring \(R\), for which a multiplication with elements of the coefficient ring of ratfun are defined. In this situation the identity element of \(R\) may be given by a further argument one which will be used for \(x^0\) for any specialized indeterminate \(x\).

The second version takes an univariate rational function and specializes the value of its indeterminate to val. Again, an optional argument one may be given.

gap> Value(x*y+y+x^7,[x,y],[5,7]);
78167

Note that the default values for one can lead to different results than one would expect: For example for a matrix \(M\), the values \(M+M^0\) and \(M+1\) are different. As Value defaults to the one of the coefficient ring, when evaluating matrices in polynomials always the correct one should be given!

66.8 Minimal Polynomials

66.8-1 MinimalPolynomial
‣ MinimalPolynomial( R, elm[, ind] )( operation )

returns the minimal polynomial of elm over the ring R, expressed in the indeterminate number ind. If ind is not given, it defaults to 1.

The minimal polynomial is the monic polynomial of smallest degree with coefficients in R that has value zero at elm.

gap> MinimalPolynomial(Rationals,[[2,0],[0,2]]);
x-2

66.9 Cyclotomic Polynomials

66.9-1 CyclotomicPolynomial
‣ CyclotomicPolynomial( F, n )( function )

is the n-th cyclotomic polynomial over the ring F.

gap> CyclotomicPolynomial(Rationals,5);
x^4+x^3+x^2+x+1

66.10 Polynomial Factorization

At the moment GAP provides only methods to factorize polynomials over finite fields (see Chapter 59), over subfields of cyclotomic fields (see Chapter 60), and over algebraic extensions of these (see Chapter 67).

66.10-1 Factors
‣ Factors( [R, ]poly[, opt] )( method )

returns a list of the irreducible factors of the polynomial poly in the polynomial ring R. (That is factors over the CoefficientsRing (66.15-3) value of R.)

For univariate factorizations, it is possible to pass a record opt as a third argument. This record can contain the following components:

onlydegs

is a set of positive integers. The factorization assumes that all irreducible factors have a degree in this set.

stopdegs

is a set of positive integers. The factorization will stop once a factor of degree in stopdegs has been found and will return the factorization found so far.

gap> f:= CyclotomicPolynomial( GF(2), 7 );
x_1^6+x_1^5+x_1^4+x_1^3+x_1^2+x_1+Z(2)^0
gap> Factors( f );
[ x_1^3+x_1+Z(2)^0, x_1^3+x_1^2+Z(2)^0 ]
gap> Factors( PolynomialRing( GF(8) ), f );
[ 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, 
  x_1+Z(2^3)^6 ]
gap> f:= MinimalPolynomial( Rationals, E(4) );
x^2+1
gap> Factors( f );
[ x^2+1 ]
gap> Factors( PolynomialRing( Rationals ), f );
[ x^2+1 ]
gap> Factors( PolynomialRing( CF(4) ), f );
[ x+(-E(4)), x+E(4) ]

66.10-2 FactorsSquarefree
‣ FactorsSquarefree( pring, upol, opt )( operation )

returns a factorization of the squarefree, monic, univariate polynomial upol in the polynomial ring pring; opt must be a (possibly empty) record of options. upol must not have zero as a root. This function is used by the factoring algorithms.

The current method for multivariate factorization reduces to univariate factorization by use of a reduction homomorphism of the form \(f(x_1,x_2,x_3) \mapsto f(x,x^p,x^{{p^2}})\). It can be very time intensive for larger degrees.

gap> Factors(x^10-y^10);
[ 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 ]

66.11 Polynomials over the Rationals

The following functions are only available to polynomials with rational coefficients:

66.11-1 PrimitivePolynomial
‣ PrimitivePolynomial( f )( operation )

takes a polynomial f 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 [newpol,coeff] with rational coeff such that coeff*newpol=f.

66.11-2 PolynomialModP
‣ PolynomialModP( pol, p )( function )

for a rational polynomial pol this function returns a polynomial over the field with p elements, obtained by reducing the coefficients modulo p.

66.11-3 GaloisType
‣ GaloisType( f )( attribute )

Let f be an irreducible polynomial with rational coefficients. This function returns the type of Gal(f) (considered as a transitive permutation group of the roots of f). It returns a number i if Gal(f) is permutation isomorphic to TransitiveGroup(n,i) where n is the degree of f.

Identification 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.

For 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.

This function requires the transitive groups library to be installed (see 50.6).

66.11-4 ProbabilityShapes
‣ ProbabilityShapes( f )( function )

Let f be an irreducible polynomial with rational coefficients. This function returns a list of the most likely type(s) of Gal(f) (see GaloisType (66.11-3)), based on factorization modulo a set of primes. It is very fast, but the result is only probabilistic.

This function requires the transitive groups library to be installed (see 50.6).

gap> f:=x^9-9*x^7+27*x^5-39*x^3+36*x-8;;
gap> GaloisType(f);
25
gap> TransitiveGroup(9,25);
[1/2.S(3)^3]3
gap> ProbabilityShapes(f);
[ 25 ]

66.12 Factorization of Polynomials over the Rationals

The following operations are used by GAP inside the factorization algorithm but might be of interest also in other contexts.

66.12-1 BombieriNorm
‣ BombieriNorm( pol )( function )

computes weighted Norm [pol]\(_2\) of pol which is a good measure for factor coefficients (see [BTW93]).

66.12-2 MinimizedBombieriNorm
‣ MinimizedBombieriNorm( f )( attribute )

This function applies linear Tschirnhaus transformations (\(x \mapsto x + i\)) to the polynomial f, trying to get the Bombieri norm of f small. It returns a list [new_polynomial, i_of_transformation].

66.12-3 HenselBound
‣ HenselBound( pol[, minpol, den] )( function )

returns the Hensel bound of the polynomial pol. If the computation takes place over an algebraic extension, then the minimal polynomial minpol and denominator den must be given.

66.12-4 OneFactorBound
‣ OneFactorBound( pol )( function )

returns the coefficient bound for a single factor of the rational polynomial pol.

66.13 Laurent Polynomials

A univariate polynomial can be written in the form \(r_0 + r_1 x + \cdots + r_n x^n\), with \(r_i \in R\). Formally, there is no reason to start with 0, if \(m\) is an integer, we can consider objects of the form \(r_m x^m + r_{{m+1}} x^{{m+1}} + \cdots + r_n x^n\). We call these Laurent polynomials. 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).

66.13-1 LaurentPolynomialByCoefficients
‣ LaurentPolynomialByCoefficients( fam, cofs, val[, ind] )( operation )

constructs a Laurent polynomial over the coefficients family fam and in the indeterminate ind (defaulting to 1) with the coefficients given by coefs and valuation val.

66.13-2 CoefficientsOfLaurentPolynomial
‣ CoefficientsOfLaurentPolynomial( laurent )( attribute )

For a Laurent polynomial laurent, this function returns a pair [cof, val], consisting of the coefficient list (in ascending order) cof and the valuation val of laurent.

gap> p:=LaurentPolynomialByCoefficients(FamilyObj(1),
> [1,2,3,4,5],-2);
5*x^2+4*x+3+2*x^-1+x^-2
gap> NumeratorOfRationalFunction(p);DenominatorOfRationalFunction(p);
5*x^4+4*x^3+3*x^2+2*x+1
x^2
gap> CoefficientsOfLaurentPolynomial(p*p);
[ [ 1, 4, 10, 20, 35, 44, 46, 40, 25 ], -4 ]

66.13-3 IndeterminateNumberOfLaurentPolynomial
‣ IndeterminateNumberOfLaurentPolynomial( pol )( attribute )

Is a synonym for IndeterminateNumberOfUnivariateRationalFunction (66.1-2).

66.14 Univariate Rational Functions

66.14-1 UnivariateRationalFunctionByCoefficients
‣ UnivariateRationalFunctionByCoefficients( fam, ncof, dcof, val[, ind] )( operation )

constructs a univariate rational function over the coefficients family fam and in the indeterminate ind (defaulting to 1) with numerator and denominator coefficients given by ncof and dcof and valuation val.

66.15 Polynomial Rings and Function Fields

While polynomials depend only on the family of the coefficients, polynomial rings \(A\) are defined over a base ring \(R\). A polynomial is an element of \(A\) if and only if all its coefficients are contained in \(R\). Besides providing domains and an easy way to create polynomials, polynomial rings can affect the behavior of operations like factorization into irreducibles.

If 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 66.1 for details about the internal numbering): Either, first create the ring and then get the indeterminates with IndeterminatesOfPolynomialRing (66.15-2).

gap> r := PolynomialRing(Rationals,["a","b"]);;
gap> indets := IndeterminatesOfPolynomialRing(r);;
gap> a := indets[1]; a := indets[2];
a
b

Alternatively, first create the indeterminates and then create the ring including these indeterminates.

gap> a:=Indeterminate(Rationals,"a":old);;
gap> b:=Indeterminate(Rationals,"b":old);;
gap> PolynomialRing(Rationals,[a,b]);;

As a convenient shortcut, intended mainly for interactive working, the \(i\)-th indeterminate of a polynomial ring \(R\) can be accessed as \(R.i\), which corresponds exactly to IndeterminatesOfPolynomialRing\(( R )[i]\) or, if it has the name nam, as \(R\).nam. Note that the number \(i\) is in general not the indeterminate number, but simply an index into the indeterminates list of \(R\).

gap> r := PolynomialRing(Rationals, ["a", "b"]:old );;
gap> r.1; r.2; r.a; r.b;
a
b
a
b
gap> IndeterminateNumberOfLaurentPolynomial(r.1);
3

Polynomials as GAP 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.)

66.15-1 PolynomialRing
‣ PolynomialRing( R, rank[, avoid] )( operation )
‣ PolynomialRing( R, names[, avoid] )( operation )
‣ PolynomialRing( R, indets )( operation )
‣ PolynomialRing( R, indetnums )( operation )

creates a polynomial ring over the ring R. If a positive integer rank is given, this creates the polynomial ring in rank indeterminates. These indeterminates will have the internal index numbers 1 to rank. The second usage takes a list names of strings and returns a polynomial ring in indeterminates labelled by names. These indeterminates have "new" internal index numbers as if they had been created by calls to Indeterminate (66.1-1). (If the argument avoid 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 indets is given. This creates the polynomial ring in the indeterminates indets. Finally, the fourth version specifies indeterminates by their index numbers.

To get the indeterminates of a polynomial ring use IndeterminatesOfPolynomialRing (66.15-2). (Indeterminates created independently with Indeterminate (66.1-1) will usually differ, though they might be given the same name and display identically, see Section 66.1.)

66.15-2 IndeterminatesOfPolynomialRing
‣ IndeterminatesOfPolynomialRing( pring )( attribute )
‣ IndeterminatesOfFunctionField( ffield )( attribute )

returns a list of the indeterminates of the polynomial ring pring, respectively the function field ffield.

66.15-3 CoefficientsRing
‣ CoefficientsRing( pring )( attribute )

returns the ring of coefficients of the polynomial ring pring, that is the ring over which pring was defined.

gap> r:=PolynomialRing(GF(7));
GF(7)[x_1]
gap> r:=PolynomialRing(GF(7),3);
GF(7)[x_1,x_2,x_3]
gap> IndeterminatesOfPolynomialRing(r);
[ x_1, x_2, x_3 ]
gap> r2:=PolynomialRing(GF(7),[5,7,12]);
GF(7)[x_5,x_7,x_12]
gap> CoefficientsRing(r);
GF(7)
gap> r:=PolynomialRing(GF(7),3);
GF(7)[x_1,x_2,x_3]
gap> r2:=PolynomialRing(GF(7),3,IndeterminatesOfPolynomialRing(r));
GF(7)[x_4,x_5,x_6]
gap> r:=PolynomialRing(GF(7),["x","y","z","z2"]);
GF(7)[x,y,z,z2]

66.15-4 IsPolynomialRing
‣ IsPolynomialRing( pring )( category )

is the category of polynomial rings

66.15-5 IsFiniteFieldPolynomialRing
‣ IsFiniteFieldPolynomialRing( pring )( category )

is the category of polynomial rings over a finite field (see Chapter 59).

66.15-6 IsAbelianNumberFieldPolynomialRing
‣ IsAbelianNumberFieldPolynomialRing( pring )( category )

is the category of polynomial rings over a field of cyclotomics (see the chapters 18 and 60).

66.15-7 IsRationalsPolynomialRing
‣ IsRationalsPolynomialRing( pring )( category )

is the category of polynomial rings over the rationals (see Chapter 17).

gap> r := PolynomialRing(Rationals, ["a", "b"] );;
gap> IsPolynomialRing(r);
true
gap> IsFiniteFieldPolynomialRing(r);
false
gap> IsRationalsPolynomialRing(r);
true

66.15-8 FunctionField
‣ FunctionField( R, rank[, avoid] )( operation )
‣ FunctionField( R, names[, avoid] )( operation )
‣ FunctionField( R, indets )( operation )
‣ FunctionField( R, indetnums )( operation )

creates a function field over the integral ring R. If a positive integer rank is given, this creates the function field in rank indeterminates. These indeterminates will have the internal index numbers 1 to rank. The second usage takes a list names of strings and returns a function field in indeterminates labelled by names. These indeterminates have "new" internal index numbers as if they had been created by calls to Indeterminate (66.1-1). (If the argument avoid 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 indets is given. This creates the function field in the indeterminates indets. Finally, the fourth version specifies indeterminates by their index number.

To get the indeterminates of a function field use IndeterminatesOfFunctionField (66.15-2). (Indeterminates created independently with Indeterminate (66.1-1) will usually differ, though they might be given the same name and display identically, see Section 66.1.)

66.15-9 IsFunctionField
‣ IsFunctionField( ffield )( category )

is the category of function fields

66.16 Univariate Polynomial Rings

66.16-1 UnivariatePolynomialRing
‣ UnivariatePolynomialRing( R[, nr] )( operation )
‣ UnivariatePolynomialRing( R[, name][, avoid] )( operation )

returns a univariate polynomial ring in the indeterminate nr over the base ring R. If nr is not given it defaults to 1.

If the number is not specified a list avoid 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 avoid.

Also a string name can be prescribed as the name of the indeterminate chosen (also avoiding the indeterminates in the list avoid if given).

66.16-2 IsUnivariatePolynomialRing
‣ IsUnivariatePolynomialRing( pring )( category )

is the category of polynomial rings with one indeterminate.

gap> r:=UnivariatePolynomialRing(Rationals,"p");
Rationals[p]
gap> r2:=PolynomialRing(Rationals,["q"]);
Rationals[q]
gap> IsUnivariatePolynomialRing(r);
true
gap> IsUnivariatePolynomialRing(r2);
true

66.17 Monomial Orderings

It 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 monomial ordering if it is total, invariant under multiplication with other monomials and admits no infinite descending chains. For details on monomial orderings see [CLO97].

In GAP, 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).

Normally the ordering chosen should be admissible, i.e. it must be compatible with products: If \(a < b\) then \(ca < cb\) for all monomials \(a, b\) and \(c\).

Each monomial ordering provides the two functions MonomialComparisonFunction (66.17-5) and MonomialExtrepComparisonFun (66.17-6) to compare monomials. These functions work as "is less than", i.e. they return true if and only if the left argument is smaller.

66.17-1 IsMonomialOrdering
‣ IsMonomialOrdering( obj )( category )

A monomial ordering is an object representing a monomial ordering. Its attributes MonomialComparisonFunction (66.17-5) and MonomialExtrepComparisonFun (66.17-6) are actual comparison functions.

66.17-2 LeadingMonomialOfPolynomial
‣ LeadingMonomialOfPolynomial( pol, ord )( operation )

returns the leading monomial (with respect to the ordering ord) of the polynomial pol.

gap> x:=Indeterminate(Rationals,"x");;
gap> y:=Indeterminate(Rationals,"y");;
gap> z:=Indeterminate(Rationals,"z");;
gap> lexord:=MonomialLexOrdering();grlexord:=MonomialGrlexOrdering();
MonomialLexOrdering()
MonomialGrlexOrdering()
gap> f:=2*x+3*y+4*z+5*x^2-6*z^2+7*y^3;          
7*y^3+5*x^2-6*z^2+2*x+3*y+4*z
gap> LeadingMonomialOfPolynomial(f,lexord);
x^2
gap> LeadingMonomialOfPolynomial(f,grlexord);
y^3

66.17-3 LeadingTermOfPolynomial
‣ LeadingTermOfPolynomial( pol, ord )( operation )

returns the leading term (with respect to the ordering ord) of the polynomial pol, i.e. the product of leading coefficient and leading monomial.

66.17-4 LeadingCoefficientOfPolynomial
‣ LeadingCoefficientOfPolynomial( pol, ord )( operation )

returns the leading coefficient (that is the coefficient of the leading monomial, see LeadingMonomialOfPolynomial (66.17-2)) of the polynomial pol.

gap> LeadingTermOfPolynomial(f,lexord);
5*x^2
gap> LeadingTermOfPolynomial(f,grlexord);
7*y^3
gap> LeadingCoefficientOfPolynomial(f,lexord);
5

66.17-5 MonomialComparisonFunction
‣ MonomialComparisonFunction( O )( attribute )

If O is an object representing a monomial ordering, this attribute returns a function that can be used to compare or sort monomials (and polynomials which will be compared by their monomials in decreasing order) in this order.

gap> MonomialComparisonFunction(lexord);
function( a, b ) ... end
gap> l:=[f,Derivative(f,x),Derivative(f,y),Derivative(f,z)];;
gap> Sort(l,MonomialComparisonFunction(lexord));l;
[ -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 ]

66.17-6 MonomialExtrepComparisonFun
‣ MonomialExtrepComparisonFun( O )( attribute )

If O is an object representing a monomial ordering, this attribute returns a function that can be used to compare or sort monomials in their external representation (as lists). This comparison variant is used inside algorithms that manipulate the external representation.

66.17-7 MonomialLexOrdering
‣ MonomialLexOrdering( [vari] )( function )

This 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.

The variables are ordered according to their (internal) index, i.e., \(x_1\) is larger than \(x_2\) and so on. If vari 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.

gap> l:=List(Tuples([1..3],3),i->x^(i[1]-1)*y^(i[2]-1)*z^(i[3]-1));
[ 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, 
  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, 
  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 ]
gap> Sort(l,MonomialComparisonFunction(MonomialLexOrdering()));l;
[ 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, 
  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, 
  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 ]
gap> Sort(l,MonomialComparisonFunction(MonomialLexOrdering([y,z,x])));l;
[ 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, 
  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, 
  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 ]
gap> Sort(l,MonomialComparisonFunction(MonomialLexOrdering([z,x,y])));l;
[ 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, 
  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, 
  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 ]

66.17-8 MonomialGrlexOrdering
‣ MonomialGrlexOrdering( [vari] )( function )

This function creates a degree/lexicographic ordering. In this ordering monomials are compared first by their total degree, then lexicographically (see MonomialLexOrdering (66.17-7)).

The variables are ordered according to their (internal) index, i.e., \(x_1\) is larger than \(x_2\) and so on. If vari 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.

66.17-9 MonomialGrevlexOrdering
‣ MonomialGrevlexOrdering( [vari] )( function )

This function creates a "grevlex" ordering. In this ordering monomials are compared first by total degree and then backwards lexicographically. (This is different than "grlex" ordering with variables reversed.)

The variables are ordered according to their (internal) index, i.e., \(x_1\) is larger than \(x_2\) and so on. If vari 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.

gap> Sort(l,MonomialComparisonFunction(MonomialGrlexOrdering()));l;
[ 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, 
  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, 
  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 ]
gap> Sort(l,MonomialComparisonFunction(MonomialGrevlexOrdering()));l;
[ 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, 
  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, 
  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 ]
gap> Sort(l,MonomialComparisonFunction(MonomialGrlexOrdering([z,y,x])));l;
[ 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, 
  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, 
  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 ]

66.17-10 EliminationOrdering
‣ EliminationOrdering( elim[, rest] )( function )

This function creates an elimination ordering for eliminating the variables in elim. Two monomials are compared first by the exponent vectors for the variables listed in elim (a lexicographic comparison with respect to the ordering indicated in elim). 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 rest, if called with two parameters).

Both elim and rest may be a list of variables or a list of variable indices.

66.17-11 PolynomialReduction
‣ PolynomialReduction( poly, gens, order )( function )

reduces the polynomial poly by the ideal generated by the polynomials in gens, using the order order of monomials. Unless gens is a Gröbner basis the result is not guaranteed to be unique.

The 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 gens.

Note that the strategy used by PolynomialReduction differs from the standard textbook reduction algorithm, which is provided by PolynomialDivisionAlgorithm (66.17-13).

66.17-12 PolynomialReducedRemainder
‣ PolynomialReducedRemainder( poly, gens, order )( function )

this operation does the same way as PolynomialReduction (66.17-11) but does not keep track of the actual quotients and returns only the remainder (it is therefore slightly faster).

66.17-13 PolynomialDivisionAlgorithm
‣ PolynomialDivisionAlgorithm( poly, gens, order )( function )

This function implements the division algorithm for multivariate polynomials as given in [CLO97, Theorem 3 in Chapter 2]. (It might be slower than PolynomialReduction (66.17-11) but the remainders are guaranteed to agree with the textbook.)

The 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 gens.

gap> bas:=[x^3*y*z,x*y^2*z,z*y*z^3+x];;
gap> pol:=x^7*z*bas[1]+y^5*bas[3]+x*z;;
gap> PolynomialReduction(pol,bas,MonomialLexOrdering()); 
[ -y*z^5, [ x^7*z, 0, y^5+z ] ]
gap> PolynomialReducedRemainder(pol,bas,MonomialLexOrdering());
-y*z^5
gap> PolynomialDivisionAlgorithm(pol,bas,MonomialLexOrdering());
[ -y*z^5, [ x^7*z, 0, y^5+z ] ]

66.17-14 MonomialExtGrlexLess
‣ MonomialExtGrlexLess( a, b )( function )

implements comparison of monomial in their external representation by a "grlex" order with \(x_1>x_2\) (This is exactly the same as the ordering by MonomialGrlexOrdering (66.17-8), see  66.17). The function takes two monomials a and b in expanded form and returns whether the first is smaller than the second. (This ordering is also used by GAP internally for representing polynomials as a linear combination of monomials.)

See section 66.21 for details on the expanded form of monomials.

66.18 Groebner Bases

A Groebner Basis of an ideal \(I\)i, in a polynomial ring \(R\), with respect to a monomial ordering, is a set of ideal generators G such that the ideal generated by the leading monomials of all polynomials in G is equal to the ideal generated by the leading monomials of all polynomials in I.

For more details on Groebner bases see [CLO97].

66.18-1 GroebnerBasis
‣ GroebnerBasis( L, O )( operation )
‣ GroebnerBasis( I, O )( operation )
‣ GroebnerBasisNC( L, O )( function )

Let O be a monomial ordering and L be a list of polynomials that generate an ideal I. This operation returns a Groebner basis of I with respect to the ordering O.

GroebnerBasisNC works like GroebnerBasis 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.

Note that GAP 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.

gap> l:=[x^2+y^2+z^2-1,x^2+z^2-y,x-y];;
gap> GroebnerBasis(l,MonomialLexOrdering());
[ x^2+y^2+z^2-1, x^2+z^2-y, x-y, -y^2-y+1, -z^2+2*y-1, 
  1/2*z^4+2*z^2-1/2 ]
gap> GroebnerBasis(l,MonomialLexOrdering([z,x,y]));
[ x^2+y^2+z^2-1, x^2+z^2-y, x-y, -y^2-y+1 ]
gap> GroebnerBasis(l,MonomialGrlexOrdering());
[ x^2+y^2+z^2-1, x^2+z^2-y, x-y, -y^2-y+1, -z^2+2*y-1 ]

66.18-2 ReducedGroebnerBasis
‣ ReducedGroebnerBasis( L, O )( operation )
‣ ReducedGroebnerBasis( I, O )( operation )

a Groebner basis \(B\) (see GroebnerBasis (66.18-1)) is reduced if no monomial in a polynomial in B is divisible by the leading monomial of another polynomial in \(B\). This operation computes a Groebner basis with respect to the monomial ordering O and then reduces it.

gap> ReducedGroebnerBasis(l,MonomialGrlexOrdering());
[ x-y, z^2-2*y+1, y^2+y-1 ]
gap> ReducedGroebnerBasis(l,MonomialLexOrdering());    
[ z^4+4*z^2-1, -1/2*z^2+y-1/2, -1/2*z^2+x-1/2 ]
gap> ReducedGroebnerBasis(l,MonomialLexOrdering([y,z,x]));
[ x^2+x-1, z^2-2*x+1, -x+y ]

For performance reasons it can be advantageous to define monomial orderings once and then to reuse them:

gap> ord:=MonomialGrlexOrdering();;
gap> GroebnerBasis(l,ord);
[ x^2+y^2+z^2-1, x^2+z^2-y, x-y, -y^2-y+1, -z^2+2*y-1 ]
gap> ReducedGroebnerBasis(l,ord);
[ x-y, z^2-2*y+1, y^2+y-1 ]

66.18-3 StoredGroebnerBasis
‣ StoredGroebnerBasis( I )( attribute )

For an ideal I in a polynomial ring, this attribute holds a list \([ B, O ]\) where \(B\) is a Groebner basis for the monomial ordering \(O\). this can be used to test membership or canonical coset representatives.

66.18-4 InfoGroebner
‣ InfoGroebner( info class )

This info class gives information about Groebner basis calculations.

66.19 Rational Function Families

All rational functions defined over a ring lie in the same family, the rational functions family over this ring.

In GAP therefore the family of a polynomial depends only on the family of the coefficients, all polynomials whose coefficients lie in the same family are "compatible".

66.19-1 RationalFunctionsFamily
‣ RationalFunctionsFamily( fam )( attribute )

creates a family containing rational functions with coefficients in fam. All elements of the RationalFunctionsFamily are rational functions (see IsRationalFunction (66.4-1)).

66.19-2 IsPolynomialFunctionsFamily
‣ IsPolynomialFunctionsFamily( obj )( category )
‣ IsRationalFunctionsFamily( obj )( category )

IsPolynomialFunctionsFamily is the category of a family of polynomials. For families over an UFD, the category becomes IsRationalFunctionsFamily (as rational functions and quotients are only provided for families over an UFD.)

gap> fam:=RationalFunctionsFamily(FamilyObj(1));
NewFamily( "RationalFunctionsFamily(...)", [ 618, 620 ],
[ 82, 85, 89, 93, 97, 100, 103, 107, 111, 618, 620 ] )

66.19-3 CoefficientsFamily
‣ CoefficientsFamily( rffam )( attribute )

If rffam has been created as RationalFunctionsFamily(cfam) this attribute holds the coefficients family cfam.

GAP does not 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.

gap> 1=Indeterminate(Rationals)^0;
false

66.20 The Representations of Rational Functions

GAP 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).

These 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 66.4 will find this out.

Each representation is associated one or several "defining attributes" that give an "external" representation (see 66.21) of the representation in the form of lists and are the defining information that tells a rational function what it is.

GAP also implements methods to compute these attributes for rational functions in other representations, provided it would be possible to express an mathematically equal 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.)

Therefore these attributes can be thought of as "conceptual" 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.

Functions 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.

Consequentially, methods for rational functions are installed for properties and not for representations.

When creating 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 "conceptual" representation in which the calculation was done is the defining attribute.

Iterated 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.

66.21 The Defining Attributes of Rational Functions

In general, rational functions are given in terms of monomials. They are represented by lists, using numbers (see 66.1) for the indeterminates.

A monomial is a product of powers of indeterminates. A monomial is stored as a list (we call this the expanded form of the monomial) of the form [inum,exp,inum,exp,...] where each inum is the number of an indeterminate and exp the corresponding exponent. The list must be sorted according to the numbers of the indeterminates. Thus for example, if \(x\), \(y\) and \(z\) are the first three indeterminates, the expanded form of the monomial \(x^5 z^8 = z^8 x^5\) is [ 1, 5, 3, 8 ]. The representation of a polynomials is a list of the form [mon,coeff,mon,coeff,...] where mon is a monomial in expanded form (that is given as list) and coeff its coefficient. The monomials must be sorted according to the total degree/lexicographic order (This is the same as given by the "grlex" monomial ordering, see MonomialGrlexOrdering (66.17-8)). We call this the external representation 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.)

The attributes that give a representation of a a rational function as a Laurent polynomial are CoefficientsOfLaurentPolynomial (66.13-2) and IndeterminateNumberOfUnivariateRationalFunction (66.1-2).

Algorithms should use only the attributes ExtRepNumeratorRatFun (66.21-2), ExtRepDenominatorRatFun (66.21-3), ExtRepPolynomialRatFun (66.21-6), CoefficientsOfLaurentPolynomial (66.13-2) and –if the univariate function is not constant– IndeterminateNumberOfUnivariateRationalFunction (66.1-2) as the low-level interface to work with a polynomial. They should not refer to the actual representation used.

66.21-1 IsRationalFunctionDefaultRep
‣ IsRationalFunctionDefaultRep( obj )( representation )

is the default representation of rational functions. A rational function in this representation is defined by the attributes ExtRepNumeratorRatFun (66.21-2) and ExtRepDenominatorRatFun (66.21-3), the values of which are external representations of polynomials.

66.21-2 ExtRepNumeratorRatFun
‣ ExtRepNumeratorRatFun( ratfun )( attribute )

returns the external representation of the numerator polynomial of the rational function ratfun. Numerator and denominator are not guaranteed to be cancelled against each other.

66.21-3 ExtRepDenominatorRatFun
‣ ExtRepDenominatorRatFun( ratfun )( attribute )

returns the external representation of the denominator polynomial of the rational function ratfun. Numerator and denominator are not guaranteed to be cancelled against each other.

66.21-4 ZeroCoefficientRatFun
‣ ZeroCoefficientRatFun( ratfun )( operation )

returns 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.

66.21-5 IsPolynomialDefaultRep
‣ IsPolynomialDefaultRep( obj )( representation )

is the default representation of polynomials. A polynomial in this representation is defined by the components and ExtRepNumeratorRatFun (66.21-2) where ExtRepNumeratorRatFun (66.21-2) is the external representation of the polynomial.

66.21-6 ExtRepPolynomialRatFun
‣ ExtRepPolynomialRatFun( polynomial )( attribute )

returns the external representation of a polynomial. The difference to ExtRepNumeratorRatFun (66.21-2) is that rational functions might know to be a polynomial but can still have a non-vanishing denominator. In this case ExtRepPolynomialRatFun has to call a quotient routine.

66.21-7 IsLaurentPolynomialDefaultRep
‣ IsLaurentPolynomialDefaultRep( obj )( representation )

This representation is used for Laurent polynomials and univariate polynomials. It represents a Laurent polynomial via the attributes CoefficientsOfLaurentPolynomial (66.13-2) and IndeterminateNumberOfLaurentPolynomial (66.13-3).

66.22 Creation of Rational Functions

The operations LaurentPolynomialByCoefficients (66.13-1), PolynomialByExtRep (66.22-2) and RationalFunctionByExtRep (66.22-1) are used to construct objects in the three basic representations for rational functions.

66.22-1 RationalFunctionByExtRep
‣ RationalFunctionByExtRep( rfam, num, den )( function )
‣ RationalFunctionByExtRepNC( rfam, num, den )( function )

constructs a rational function (in the representation IsRationalFunctionDefaultRep (66.21-1)) in the rational function family rfam, the rational function itself is given by the external representations num and den for numerator and denominator. No cancellation takes place.

The variant RationalFunctionByExtRepNC 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.

66.22-2 PolynomialByExtRep
‣ PolynomialByExtRep( rfam, extrep )( function )
‣ PolynomialByExtRepNC( rfam, extrep )( function )

constructs a polynomial (in the representation IsPolynomialDefaultRep (66.21-5)) in the rational function family rfam, the polynomial itself is given by the external representation extrep.

The variant PolynomialByExtRepNC 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.

gap> fam:=RationalFunctionsFamily(FamilyObj(1));;
gap> p:=PolynomialByExtRep(fam,[[1,2],1,[2,1,15,7],3]);
3*y*x_15^7+x^2
gap> q:=p/(p+1);
(3*y*x_15^7+x^2)/(3*y*x_15^7+x^2+1)
gap> ExtRepNumeratorRatFun(q);
[ [ 1, 2 ], 1, [ 2, 1, 15, 7 ], 3 ]
gap> ExtRepDenominatorRatFun(q);
[ [  ], 1, [ 1, 2 ], 1, [ 2, 1, 15, 7 ], 3 ]

66.22-3 LaurentPolynomialByExtRep
‣ LaurentPolynomialByExtRep( fam, cofs, val, ind )( function )
‣ LaurentPolynomialByExtRepNC( fam, cofs, val, ind )( function )

creates a Laurent polynomial in the family fam with [cofs,val] as value of CoefficientsOfLaurentPolynomial (66.13-2). 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, LaurentPolynomialByCoefficients (66.13-1) should be used.

66.23 Arithmetic for External Representations of Polynomials

The following operations are used internally to perform the arithmetic for polynomials in their "external" representation (see 66.21) as lists.

Functions to perform arithmetic with the coefficient lists of Laurent polynomials are described in Section 23.4.

66.23-1 ZippedSum
‣ ZippedSum( z1, z2, czero, funcs )( operation )

computes the sum of two external representations of polynomials z1 and z2. czero is the appropriate coefficient zero and funcs a list [ monomial_less, coefficient_sum ] containing a monomial comparison and a coefficient addition function. This list can be found in the component fam!.zippedSum of the rational functions family.

Note that coefficient_sum must be a proper "summation" function, not a function computing differences.

66.23-2 ZippedProduct
‣ ZippedProduct( z1, z2, czero, funcs )( operation )

computes the product of two external representations of polynomials z1 and z2. czero is the appropriate coefficient zero and funcs a list [ monomial_prod, monomial_less, coefficient_sum, coefficient_prod] containing functions to multiply and compare monomials, to add and to multiply coefficients. This list can be found in the component fam!.zippedProduct of the rational functions family.

66.23-3 QuotientPolynomialsExtRep
‣ QuotientPolynomialsExtRep( fam, a, b )( function )

Let a and b the external representations of two polynomials in the rational functions family fam. This function computes the external representation of the quotient of both polynomials, it returns fail if the polynomial described by b does not divide the polynomial described by a.

66.24 Cancellation Tests for Rational Functions

The operation Gcd (56.7-1) 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

66.24-1 RationalFunctionByExtRepWithCancellation
‣ RationalFunctionByExtRepWithCancellation( rfam, num, den )( function )

constructs a rational function as RationalFunctionByExtRep (66.22-1) does but tries to cancel out common factors of numerator and denominator, calling TryGcdCancelExtRepPolynomials (66.24-2).

66.24-2 TryGcdCancelExtRepPolynomials
‣ TryGcdCancelExtRepPolynomials( fam, a, b )( function )

Let a and b be the external representations of two polynomials. This function tries to cancel common factors between the corresponding polynomials and returns a list \([ a', b' ]\) of external representations of cancelled polynomials. As there is no proper multivariate GCD cancellation is not guaranteed to be optimal.

66.24-3 HeuristicCancelPolynomialsExtRep
‣ HeuristicCancelPolynomialsExtRep( fam, ext1, ext2 )( operation )

is called by TryGcdCancelExtRepPolynomials (66.24-2) to perform the actual work. It will return either fail or a new list of of external representations of cancelled polynomials. The cancellation performed is not necessarily optimal.

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