A Lower Bound for the Determinantal Complexity of a Hypersurface

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 × 3 permanent is 7. We also prove that for n> 3 , there is no nonsingular hypersurface in Pⁿ of degree d that has an expression as a determinant of a d× d matrix of linear forms, while on the other hand for 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.