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.