The Exact Reach of Conormal Invariants in Determinantal Complexity: a Quadratic No-Go Theorem (opens in new tab)

We study the polar (conormal) method for determinantal-complexity lower bounds, including the framework used in the companion bound dc(sum_i x_i^N) >= (1/(4e)-o(1))N^2. We obtain quantitative results on both sides of the method: the intersection-theoretic complexity of kernel-incidence constructions and the size of the characteristic-cycle invariants they can detect. For a size-m determinantal representation in N variables, we identify the coran...