Computation of the Hilbert Series for the Support-Minors Modeling of the MinRank Problem

TL;DR

This paper derives a formula and proof for the complete Hilbert Series of the Support-Minors modeling of the MinRank problem, advancing algebraic understanding of this computational problem.

Contribution

It provides the first exact formula for the Hilbert Series of the Support-Minors modeling, extending previous heuristic and partial results.

Findings

Complete Hilbert Series formula for Support-Minors modeling

Extension of determinantal ideal results to specific minors

Enhanced algebraic analysis of the MinRank problem

Abstract

The MinRank problem is a simple linear algebra problem: given matrices with coefficients in a field, find a non trivial linear combination of the matrices that has a small rank. There are several algebraic modeling of the problem. The main ones are: the Kipnis-Shamir modeling, the Minors modeling and the Support-Minors modeling. The Minors modeling has been studied by Faug{\`e}re et al. in 2010, where the authors provide an analysis of the complexity of computing a Gr{\"o}bner basis of the modeling, through the computation of the exact Hilbert Series for a generic instance. For the Support-Minors modeling, the first terms of the Hilbert Series are given by Bardet et al. in 2020 based on an heuristic and experimental work. In this work, we provide a formula and a proof for the complete Hilbert Series of the Support Minors modeling for generic instances. This is done by adapting well known results on determinantal ideals to an ideal generated by a particular subset of the set of all minors of a matrix of variables. We then show that this ideal is generated by