The symmetric determinantal complexity sdc(f) of a polynomial f is the least m such that f = det(M) for an m x m symmetric matrix M of affine-linear forms. We prove, over the complex numbers, that sdc(sum_{i=1}^n x_i^n) >= (1/(2e) - o(1)) n^2. This is a symmetric companion to the author's non-symmetric polar-degree preprint (arXiv:7680505); the method parallels that work, but the proof here is self-contained and redoes the load-bearing local incidence analysis in the symmetric setting. The general theorem: if X = V(f) in P^{N-1} is a smooth degree-d hypersurface, N >= 3, and f = det(A_0 + sum x_i A_i) with all A_i symmetric of size m, then the top polar degree d(d-1)^{N-2} is at most 2^{N-2} C(m, N-1). The proof uses the symmetric rank-one kernel incidence M(z,x) u = 0. At a genuine polar point M has rank m-1, and a symmetric Schur-complement normal form eliminates the unique kernel line scheme-theoretically; on the resulting local graph the lifted conormal forms u^T A_i u are a common unit multiple of the partials d_i f, so the lifted polar equations cut the ordinary polar slice up to units and each genuine lifted polar point is a zero-dimensional isolated solution. Multihomogeneous Bezout on P^N x P^{m-1} then yields the bound 2^{N-2} C(m, N-1). For F_n = sum x_i^n this gives the constant 1/(2e). More generally, for F_{N,d} = sum_{i=1}^N x_i^d the same theorem gives sdc(F_{N,d}) >= (1/(2e) - o_N(1)) N(d-1) as N -> infinity. We give an explicit symmetric representation of F_{N,d} of size 2N(d+1)+1, so the diagonal bounds are non-vacuous and tight up to a constant. The result is for exact symmetric determinantal complexity in characteristic zero; it is not a border statement and not a uniform positive-characteristic theorem.

翻译：多项式f的对称行列式复杂度sdc(f)定义为满足f = det(M)的最小整数m，其中M为m×m仿射线性型对称矩阵。本文在复数域上证明：sdc(∑_{i=1}^n x_i^n) ≥ (1/(2e) - o(1)) n²。这是作者非对称极次预印本(arXiv:7680505)的对称版本；方法与该工作类似，但本文证明自包含且重新处理了对称框架下的承重局部相交分析。一般性定理：设X = V(f) ⊂ P^{N-1}为光滑d次超曲面，N ≥ 3，且f = det(A_0 + ∑ x_i A_i)且所有A_i为m阶对称矩阵，则最高极次d(d-1)^{N-2} ≤ 2^{N-2} C(m, N-1)。证明利用对称秩一核入射M(z,x) u = 0。在真型极点上M的秩为m-1，对称Schur补标准型在概形意义下消去唯一核直线；在所得局部图上，提升余法型u^T A_i u是偏导数∂_i f的公共单位倍数，因此提升极方程在单位元意义下截取常义极切片，且每个真型提升极点为零维孤立解。对P^N × P^{m-1}运用多齐次Bezout定理得到界限2^{N-2} C(m, N-1)。对于F_n = ∑ x_i^n可得常数1/(2e)。更一般地，对于F_{N,d} = ∑_{i=1}^N x_i^d，当N → ∞时相同定理给出sdc(F_{N,d}) ≥ (1/(2e) - o_N(1)) N(d-1)。我们给出F_{N,d}在2N(d+1)+1阶上的显式对称表示，因此对角界限非平凡且紧致至常数因子。该结果针对特征零域上的精确对称行列式复杂度，并非边界声明亦非一致正特征定理。