A near-quadratic lower bound on the border determinantal complexity of ${\sum }_i{x}_i^n$ via conormal specialization

Abstract

The border determinantal complexity $\dcb(f)$ of a polynomial $f$ is the least $m$ such that $f$ is a limit of determinants of $m\times m$ matrices of affine-linear forms. We prove that for every $n\ge 3$, over $\CC$, \[ \dcb\Big(\sum_{i=1}^n x_i^n\Big)\ \ge\ \frac{(n-1)^2}{4e}, \qquad \sdcb\Big(\sum_{i=1}^n x_i^n\Big)\ \ge\ \frac{(n-1)^2}{2e} \] in the ordinary and symmetric models respectively; both match the known $O(n^2)$ upper bounds up to the constant. To our knowledge these are the first border determinantal lower bounds for an explicit family that are superlinear in the number of variables: the known quadratic border bound for the permanent reads the \emph{dimension} of the dual variety and is linear in its number of variables, whereas we transfer the dual \emph{degree}. The proof has two ingredients. The first is an unconditional bound on the slot-$(n-2)$ conormal multidegree of the multiplicity-one Gauss-graph cycle of an arbitrary affine-linear determinant -- singular, reducible, and non-reduced fibers allowed -- by a multihomogeneous Bézout count of a lifted kernel incidence. The second is a specialization argument: along any degeneration $\det A_c\to {\sum }_i{x}_i^n$, the flat limit of these Gauss-graph cycles contains the conormal variety of the Fermat cone with positive coefficient. A cone-shift identity converts that conormal multidegree into the classical dual degree $n(n-1{)}^{n-2}$ of the smooth Fermat hypersurface, and an $(n-1)$-st root yields the quadratic bound. The exact lower bounds of the author's companion manuscripts follow as corollaries.