A lower bound for the determinantal complexity of a hypersurface

TL;DR

This paper establishes lower bounds on the determinantal complexity of hypersurfaces, providing exact values for specific cases like the 3x3 permanent and addressing a question about cubic surfaces.

Contribution

It introduces new lower bounds for determinantal complexity based on singular locus codimension and solves an open problem regarding the complexity of a singular cubic surface.

Findings

Determinantal complexity of degree d hypersurfaces is at least codimension of singular locus plus one for codimension ≥ 5.

Determinantal complexity of 3x3 permanent is exactly 7.

The unique singular cubic surface containing a single line has determinantal complexity 5.

Abstract

We prove that the determinantal complexity of a hypersurface of degree $d > 2$ is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least $5$. As a result, we obtain that the determinantal complexity of the $3 \times 3$ permanent is $7$. We also prove that for $n> 3$, there is no nonsingular hypersurface in $\mathbf{P}^n$ of degree $d$ that has an expression as a determinant of a $d \times d$ matrix of linear forms while on the other hand for $n \le 3$, a general determinantal expression is nonsingular. Finally, we answer a question of Ressayre by showing that the determinantal complexity of the unique (singular) cubic surface containing a single line is $5$.