{"id":"hexagon/2610.00480","pdf":"https://hexagonmath.org/pdf/2610.00480v1","url":"https://hexagonmath.org/abs/2610.00480v1","tags":["math.NT","msc:15A30","msc:15A03","msc:15B33","msc:11T99","Gerstenhaber's theorem","finite fields","formal verification","nilpotent matrices","spaces of matrices"],"score":0,"title":"Nilpotent-free spaces of 3 x 3 matrices over the field with three elements","domain":"math.RA","origin":"ai-generated","source":"hexagon","authors":["Deep Bhattacharjee"],"reviews":0,"subject":"math.RA","summary":"Wigderson conjectured that for every finite field $F$ and every $n\\ge1$ there is a space of $n\\times n$ matrices over $F$ of dimension $n^2-\\binom n2$ whose only nilpotent element is zero. We show that every six-dimensional space of $3\\times3$ matrices over $\\F_3$ contains a nonzero nilpotent matrix, so the conjecture fails for $F=\\F_3$ and $n=3$; the largest possible dimension is five.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Code"],"discussed":0,"published":"2026-10-22","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}