kronecker

Functions related to computing the Kronecker representation of a system of polynomials.

sage_acsv.kronecker.kronecker(system, vs, linear_form=None)[source]
sage_acsv.kronecker.kronecker_representation(system, vs, linear_form=None)[source]

Computes the Kronecker Representation of a system of polynomials.

INPUT:

  • system – A system of polynomials in \(d\) variables defining a zero-dimensional (finite) variety

  • vs – Variables of the system

  • linear_form – (Optional) A linear combination of the input variables that separates the critical point solutions

OUTPUT:

A polynomial \(P\) and \(d\) polynomials \(Q1, ..., Q_d\) such that \(z_i = Q_i(u)/P'(u)\) for \(u\) ranging over the roots of \(P\).

EXAMPLES:

sage: from sage_acsv.kronecker import kronecker_representation
sage: var('x,y')
(x, y)
sage: kronecker_representation([x**3+y**3-10, y**2-2], [x,y], x+y)
(u_^6 - 6*u_^4 - 20*u_^3 + 36*u_^2 - 120*u_ + 100,
 [60*u_^3 - 72*u_^2 + 360*u_ - 600, 12*u_^4 - 72*u_^2 + 240*u_])