A lower bound for the determinantal complexity of a hypersurface

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 $5$.

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