This section briefly describes how to use the implementation of the main algorithm designed by B. Bank, M. Giusti, J. Heintz, G. Lecerf, G. Matera, and P. Solerno in the article Degeneracy loci and polynomial equation solving.

Definition

Let , be polynomials of satisfying the following conditions:

• form a reduced regular sequence outside of , and

• is a smooth quasi-affine variety.

Let , and let for and , be a matrix of polynomials in , with .

For , we denote by the rank of the complex -matrix , and we let . For any matrix

with , we write

and define

The function degeneracy_locus% computes with a high probability of success.

Use from the interpreter

Mmx]  

use "geomsolvex";
type_mode? := true;

Mmx]  

X == coordinate ('x); Y == coordinate ('y); Z == coordinate ('z);

Mmx]  

x == polynomial_dag (1 :> Rational, X);
y == polynomial_dag (1 :> Rational, Y);
z == polynomial_dag (1 :> Rational, Z);

Mmx]  

P == x^2 + y^2 + z^2 - 1

Mmx]  

G == [ P ]

Mmx]  

H == [] :> Vector(Polynomial_dag(Rational))

Mmx]  

F == matrix (derive (P, X), derive (P, Y), derive (P, Z))

Mmx]  

a == [1 :> Rational, 2, 3; 3, 5, 2 ]

Mmx]  

degeneracy_locus% (G, H, F, a)