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 $$3 \times 3$$ permanent is 7. We also prove that for $$n> 3$$ , there is no nonsingular hypersurface in $${\mathbb {P}}^n$$ 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. [ABSTRACT FROM AUTHOR]