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

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{è}re et al. in 2010, where the authors provide an analysis of the complexity of computing a Gr{ö}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