Short polynomials in determinantal ideals

TL;DR

This paper proves that determinantal ideals generated by t-minors cannot contain polynomials with t!/2 or fewer terms, implying such polynomials must have more than t!/2 terms if they vanish on matrices of rank less than t.

Contribution

It establishes a lower bound on the number of terms in polynomials vanishing on low-rank matrices within determinantal ideals, revealing a new combinatorial property.

Findings

Polynomials vanishing on rank-deficient matrices have more than t!/2 terms.

Determinantal ideals do not contain polynomials with t!/2 or fewer terms.

Provides a geometric interpretation of the algebraic result.

Abstract

We show that a determinantal ideal generated by $t$-minors does not contain any nonzero polynomials with $t!/2$ or fewer terms. Geometrically this means that any nonzero polynomial vanishing on all matrices of rank at most $t-1$ has more than $t!/2$ terms.