A Lower Bound for the Determinantal Complexity of a Hypersurface

Abstract

We prove that the determinantal complexity of a hypersurface of degree $$d > 2$$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 $$3 \times 3$$3×3 permanent is 7. We also prove that for $$n> 3$$n>3, there is no nonsingular hypersurface in $${\mathbb {P}}^n$$Pn of degree d that has an expression as a determinant of a $$d \times d$$d×d matrix of linear forms, while on the other hand for $$n \le 3$$n≤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.