The best known lower bound on the determinantal complexity of the $4\times4$ permanent is 10. In this short note, I prove that this value is exactly 15: the $4\times4$ permanent is the determinant of a $15\times15$ matrix of affine linear forms, and cannot be that of any smaller matrix. To ... more >>>

OpenAI's Sep 2026 preprint introduced the first cubic lower bound on the border DC of the permanent. Their preprint generalizes the method I introduced in my June 2026 preprint (https://arxiv.org/abs/2606.15970; see also TR26-275). However, their proof only applies over $\mathbb{C}$.

In this note, I prove that the cubic lower bound ... more >>>

In an earlier note (arXiv:2606.11090) I showed that the border determinantal complexity of the power sum x_1^d + ... + x_N^d is ?(Nd). The proof counted the simple solutions of an explicit polynomial system and compared that number with what a small determinant can produce. This left two questions. Does ... more >>>

I give a short proof that the border determinantal complexity of F_n = x_1^n + ... + x_n^n over 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 + ... + x_N^d is Theta(Nd) for ... more >>>

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 ... more >>>