{
  "$type": "site.standard.document",
  "bskyPostRef": {
    "cid": "bafyreia6xucsqbp23qmercz4rzyunecz6mfx5gtvbusjmcnygzr4ghwz3e",
    "uri": "at://did:plc:3fychdutjjusoqeq24ljch6q/app.bsky.feed.post/3mnwa67gxzwi2"
  },
  "path": "/report/2026/097",
  "publishedAt": "2026-06-09T17:39:34.000Z",
  "site": "https://eccc.weizmann.ac.il",
  "textContent": "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.",
  "title": "TR26-097 |  A symmetric determinantal lower bound for diagonal power sums\\\\ via polar degree | \n\n\tKarthik Sheshadri"
}