TR26-097 | A symmetric determinantal lower bound for diagonal power sums\\ via polar degree | Karthik Sheshadri

The symmetric determinantal complexity $\sdc(f)$ of a polynomial $f$ is the least $m$ such that $f=\Det(M)$ for an $m\times m$ symmetric matrix $M$ of affine-linear forms. We prove, over $\CC$, that \[ \sdc\!\left(\sum_{i=1}^n x_i^n\right) \ge \left(\frac{1}{2e}-o(1)\right)n^2 . \] The result is a symmetric companion to the author's non-symmetric polar-degree preprint~\cite{SheshadriArxiv}. The method parallels that work, but the proof below is self-contained and redoes the load-bearing local incidence analysis in the symmetric setting. The general theorem is the following. If $X=V(f)\subset\PP^{N-1}$ is a smooth degree-$d$ hypersurface, $N\ge3$, and $f=\Det(A_0+\sum_{i=1}^N x_iA_i)$ with all $A_i$ symmetric of size $m$, then \[ \pdeg_{\mathrm{top}}(X)=d(d-1)^{N-2} \le 2^{N-2}\binom{m}{N-1}. \] The proof uses the symmetric rank-one kernel incidence $\Mcal(z,x)u=0$, where $\Mcal=zA_0+\sum_i x_iA_i$. At a genuine polar point, $\Mcal$ has rank $m-1$, and the symmetric local normal form \[ \Mcal=\begin{pmatrix}B&c\\ c^{\mathsf T}&s\end{pmatrix},\qquad \det B\in\OO^\times, \] eliminates the unique projective kernel line scheme-theoretically: $u=(-B^{-1}c,1)$ and $\det\Mcal=(\det B)(s-c^{\mathsf T}B^{-1}c)$. On this local graph, $\adj(\Mcal)=(\det B)uu^{\mathsf T}$ along the determinant hypersurface, so the lifted conormal forms $u^{\mathsf T}A_i u$ are a common unit multiple of the ordinary partial derivatives $\partial_i f$. Hence the lifted polar equations cut the ordinary polar slice, up to units, and every genuine lifted polar point is a zero-dimensional scheme-theoretic isolated solution. Multihomogeneous Bezout on $\PP^N_{[z:x]}\times\PP^{m-1}{[u]}$ then gives \[ [H^N U^{m-1}],H(H+U)^m(2U)^{N-2} =2^{N-2}\binom{m}{N-1}. \] For $F_n=\sum_i x_i^n$ this bounds $n(n-1)^{n-2}$ and yields the stated constant $1/(2e)$. More generally, for $F{N,d}=\sum_{i=1}^N x_i^d$ the same theorem gives $\sdc(F_{N,d})\ge(1/(2e)-o_N(1))N(d-1)$ as $N\to\infty$, uniformly for $d\ge2$. We also give an explicit symmetric determinantal representation of $F_{N,d}$ of size $2N(d+1)+1$, showing that the diagonal lower bounds are non-vacuous and tight up to a constant factor. The result is for exact symmetric determinantal complexity in characteristic zero; it is not a border-complexity statement and it is not a uniform positive-characteristic theorem.