The research proves that VPnc is closed under taking factors over every field.

Factor Closure in Noncommutative Algebraic Complexity https://arxiv.org/abs/2611.02039v1 Authors: V. Arvind, Pushkar S. JoglekarWe 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 outer factors $a,b$, has a circuit of size $O(sD^6)$. The pro

TR26-300 | Factor Closure in Noncommutative Algebraic Complexity | Pushkar Joglekar, Vikraman Arvind https://eccc.weizmann.ac.il/report/2026/300 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 outer factors $a,b$, has a circuit of size