No short polynomials vanish on bounded rank matrices

TL;DR

This paper proves that the simplest nonzero polynomials vanishing on bounded-rank matrices are determinants and Pfaffians, showing their fundamental role in the algebraic structure of these matrix varieties.

Contribution

It establishes that determinants and Pfaffians are the minimal-term polynomials defining rank conditions, and characterizes the structure of ideals generated by minors and Pfaffians.

Findings

Determinants and Pfaffians are the shortest nonzero polynomials vanishing on bounded-rank matrices.

In the ideal generated by minors or Pfaffians, no polynomial with fewer terms exists than these determinants or Pfaffians.

The ideal of a general t-dimensional subspace of affine space contains no polynomials with fewer than t+1 terms.

Abstract

We show that the shortest nonzero polynomials vanishing on bounded-rank matrices and skew-symmetric matrices are the determinants and Pfaffians characterising the rank. Algebraically, this means that in the ideal generated by all $t$-minors or $t$-Pfaffians of a generic matrix or skew-symmetric matrix one cannot find any polynomial with fewer terms than those determinants or Pfaffians, respectively, and that those determinants and Pfaffians are essentially the only polynomials in the ideal with that many terms. As a key tool of independent interest, we show that the ideal of a sufficiently general $t$-dimensional subspace of an affine $n$-space does not contain polynomials with fewer than $t+1$ terms.