We prove that $\VPnc$ is closed under taking factors over every field. If a noncommutative circuit of size $s$ computes a nonconstant polynomial of degree at most $D$, then each left or right factor has a circuit of size $O(sD^3)$. A central factor $g$ in a decomposition $f=agb$, with nonconstant ... more >>>

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 >>>