A Lower Bound for the Determinantal Complexity of a Hypersurface

We prove that the determinantal complexity of a hypersurface of degree [Formula omitted] 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 [Formula omitted] permanent is 7. We also prove that for [Formula omitted], there is no nonsingular hypersurface in [Formula omitted] of degree d that has an expression as a determinant of a [Formula omitted] matrix of linear forms, while on the other hand for [Formula omitted], 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.
Author(s): Jarod Alper [sup.1] , Tristram Bogart [sup.2] , Mauricio Velasco [sup.2] Author Affiliations: (1) 0000 0001 2180 7477grid.1001.0Mathematical Sciences Institute, Australian National University, 0200, Canberra, ACT, Australia (2) 0000000419370714grid.7247.6Departamento [...]