Andrews, Garg, and Schost (FOCS~2026) recently, in a breakthrough result, placed Hilbert's Nullstellensatz (HN) in the Counting Hierarchy ($CH$) using constant-depth circuits for multivariate resultants. While HN deals with exact polynomial satisfiability, Guo, Saxena, and Sinhababu (CCC~2018; Theory of Computing~2019) introduced approximate polynomial satisfiability (APS) as a natural approximative analogue ... more >>>

This note is an exposition of the recent preprints from OpenAI that establish the bound $\omega \leq 9/4$ on the matrix multiplication exponent. We argue that the heart of these preprints is a new technique, which we formulate as the untwisting lemma for semidirect sums of tensors. This gives a ... more >>>

We study depth-$4$ algebraic circuits with bounded top and bottom fan-in over fields of characteristic $0$. These circuits compute sums of at most $k$ products of polynomials of degree at most $\delta$. We denote this class of circuits by $\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}$. Our hard polynomial for this class is $P_{k + 1,n}$, ... more >>>