Let $\tau(k)$ be the minimum number of arithmetic operations
required to build the integer $k \in \N$ from the constant 1.
A sequence $x_k$ is said to be ``easy to compute'' if
there exists a polynomial $p$ such that $\tau(x_k) \leq p(\log k)$
for all $k \geq ... more >>>

We study preprocessing algorithms for the function-inversion problem. In this problem, an algorithm gets oracle access to a function $f\colon[N] \to [N]$ and takes as input $S$ bits of auxiliary information about $f$, along with a point $y \in [N]$. After running for time $T$, the algorithm must output an ... 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 >>>

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 >>>

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 >>>