A short proof of a quadratic lower bound on the border determinantal complexity of $x_1^n+\cdots +x_n^n$ over $\mathbb{C}$: A simple human exposition

Abstract

I give a short proof that the border determinantal complexity of $F_n=x_1^n+\cdots +x_n^n$ over $\mathbb{C}$ is at least $(n-1{)}^2/(4e)$, and at least $(n-1{)}^2/(2e)$ in the symmetric model. The same argument shows that the border determinantal complexity of $x_1^d+\cdots +x_N^d$ is $\Theta (Nd)$ for all $N\ge 3$ and $d\ge 2$. The earlier article arXiv:2606.13628, by the same author, introduced the first superlinear lower bounds and includes verification by Lean formalization. However, the AI written exposition was hard to follow. Kumar and Volk (arXiv:2609.34462) give a short proof of the exact ordinary case, but do not address the border or symmetric cases. In this article, I provide a greatly simplified exposition of the border and symmetric cases, using the implicit function theorem and an explicit section of simple solutions in place of the conormal specialization arguments of arXiv:2606.13628. I choose an explicit polynomial system with $n(n-1{)}^{n-2}$ simple solutions for $F_n$. I then use the implicit function theorem to ensure that each point persists under small perturbations. Finally, kernel equations and the multihomogeneous Bézout theorem bound the number of these points for a small determinant. The exposition is due to the human author. The mathematical simplification was obtained in discussion with ChatGPT Astra and refined in conversation with Claude Opus 5.5.

arXiv commentv2: substantially revised and retitled; supersedes v1, which proved the exact symmetric bound via polar degree. Now gives a short proof of the ordinary and symmetric border bounds