Computing images of polynomial maps

TL;DR

This paper introduces a new algebro-geometric algorithm to compute the constructible image set of polynomial maps, extending beyond closure computation, and applies it to a problem in matrix product states.

Contribution

It presents a novel algorithm for computing the actual image set of polynomial maps, not just its closure, using advanced algebraic geometry techniques.

Findings

Algorithm successfully computes constructible images of polynomial maps.

Applied method resolves a question about non-closedness of certain matrix product states.

Demonstrates practical utility in algebraic geometry and quantum information contexts.

Abstract

The image of a polynomial map is a constructible set. While computing its closure is standard in computer algebra systems, a procedure for computing the constructible set itself is not. We provide a new algorithm, based on algebro-geometric techniques, addressing this problem. We also apply these methods to answer a question of W. Hackbusch on the non-closedness of site-independent cyclic matrix product states for infinitely many parameters.