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The sign rank of a ${\pm 1}$-valued matrix $A$ is the minimum rank of a real matrix that agrees with $A$ entrywise in sign. For every sufficiently large $d$, we give an explicit matrix $A$ with $2^{\Theta(d^2)}$ rows and columns and sign rank $\Theta(d)$ such that $A \land A$ has ... more >>>
We resolve two open problems regarding the Sherali-Adams proof system. First, we show that Res(2), which manipulates 2-DNF formulas, is exponentially stronger than Sherali-Adams. This provides a sharp separation in power between Sherali-Adams and the bounded-depth Frege hierarchy, as Sherali-Adams polynomially simulates Resolution. Second, we prove that Sherali-Adams is not ... 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 >>>
