11 Algebraic Varieties
In FinInG we provide some basic functionality for algebraic varieties defined over finite fields. The algebraic varieties in FinInG are defined by a list of multivariate polynomials over a finite field, and an ambient geometry. This ambient geometry is either a projective space, and then the algebraic variety is called a projective variety, or an affine geometry, and then the algebraic variety is called an affine variety. In this chapter we give a brief overview of the features of FinInG concerning these two types of algebraic varieties.
11.1 Projective Varieties
A projective variety in FinInG is an algebraic variety in a projective space defined by a list of homogeneous polynomials over a finite field.
11.1-1 ProjectiveVariety
> ProjectiveVariety ( pg, pring, pollist ) | ( operation ) |
> ProjectiveVariety ( pg, pollist ) | ( operation ) |
> AlgebraicVariety ( pg, pring, pollist ) | ( operation ) |
> AlgebraicVariety ( pg, pollist ) | ( operation ) |
Returns: a projective algebraic variety
The argument pg is a projective space over a finite field F, the argument pring is a multivariate polynomial ring defined over (a subfield of) F, and pollist is a list of homogeneous polynomials in pring.
gap> F:=GF(9);
GF(3^2)
gap> r:=PolynomialRing(F,4);
GF(3^2)[x_1,x_2,x_3,x_4]
gap> pg:=PG(3,9);
PG(3, 9)
gap> f1:=r.1*r.3-r.2^2;
x_1*x_3-x_2^2
gap> f2:=r.4*r.1^2-r.4^3;
x_1^2*x_4-x_4^3
gap> var:=AlgebraicVariety(pg,[f1,f2]);
Algebraic Variety V( [ x_1*x_3-x_2^2, x_1^2*x_4-x_4^3 ] ) in PG(3, 9)
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11.2 Affine Varieties
An affine variety in FinInG is an algebraic variety in an affine space defined by a list of polynomials over a finite field.
11.2-1 AffineVariety
> AffineVariety ( ag, pring, pollist ) | ( operation ) |
> AffineVariety ( ag, pollist ) | ( operation ) |
> AlgebraicVariety ( ag, pring, pollist ) | ( operation ) |
> AlgebraicVariety ( ag, pollist ) | ( operation ) |
Returns: an affine algebraic variety
The argument ag is an affine space over a finite field F, the argument pring is a multivariate polynomial ring defined over (a subfield of) F, and pollist is a list of polynomials in pring.