The Exact Reach of Conormal Invariants in Determinantal Complexity: a Quadratic No-Go Theorem

Abstract

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 corank-one kernel incidence with the conormal variety of the generic determinant. An excess-one degeneracy-locus computation yields a closed formula for the associated polar intersection number T(N,m), together with rational generating functions and explicit evaluations including T(3,m)=m(m-1), T(4,m)=m(m-1)^2, and T(5,5)=220. We also compare these counts with the multihomogeneous Bezout estimates used in the companion work and establish asymptotic sharpness at the per-root scale. For an arbitrary degree-d hypersurface X in P^(N-1), possibly singular and reducible, we prove a uniform bound on the conormal multidegrees appearing in its characteristic cycle: m_S delta_i(Con(S-bar)) <= 8(d-1)^(N-1)+O(N). The proof combines bounds on generic-slice Euler characteristics, an explicit analysis of the transform from Euler-characteristic data to conormal multidegrees, and Kashiwara positivity for characteristic cycles. Similar bounds are obtained for vanishing-cycle and Milnor-class variants. Combining these results yields general upper bounds on determinantal-complexity lower bounds obtainable from characteristic-cycle invariants via kernel-corank incidence constructions. In particular, along the diagonal d=N, the resulting lower bounds are at most quadratic in N. We conclude by discussing possible extensions involving scheme-theoretic conormal information and other geometric invariants.