HEXAGONbeta

Nilpotent-free spaces of 3 x 3 matrices over the field with three elements

A mix of human-written and AI-generated textHuman understanding: some partsmath.RA — Rings and Algebrasmath.NT — Number Theory

Contributed by Deep Bhattacharjee ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

Wigderson conjectured that for every finite field FF and every n≥1n\ge1 there is a space of n×nn\times n matrices over FF of dimension n2−(n2)n^2-\binom n2 whose only nilpotent element is zero. We show that every six-dimensional space of 3×33\times3 matrices over $\F_3$ contains a nonzero nilpotent matrix, so the conjecture fails for $F=\F_3$ and n=3n=3; the largest possible dimension is five.

Provenance statement

Deep Bhattacharjee chose the problem and directed the work. Claude (Anthropic), through Claude Code, carried out the searches and computations, wrote the checking programs and the Lean 4 proofs and drafted the paper. Refutes Wigderson's Conjecture 8 for F_3, n = 3; the main step uses native_decide, rechecked by two C programs. The proofs are written out in the paper; the programs and their outputs are in the release. The author approved the final version. Code and data: release v1.0.0, https://github.com/creelie/congenial-waddle/releases/tag/v1.0.0.
AI provenance filesAI-PROVENANCE-HEXAGON.md · 1 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

References

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Show all 13 referencesShow fewer references
  1. V. N. Serezhkin, Linear transformations preserving nilpotency , Vestsī Akad. Navuk BSSR Ser. Fīz.-Mat. Navuk (1985), no. 6, 46--50 (Russian).
  2. E. Warning, Bemerkung zur vorstehenden Arbeit von Herrn Chevalley , Abh. Math. Sem. Univ. Hamburg 11 (1935), 76--83. 10.1007/BF02940715DOI
  3. Y. Wigderson, Duality for matrix space questions , preprint, 2026, to appear in Amer. Math. Monthly. arXiv:2609.29177 (https://arxiv.org/abs/2609.29177)arXiv

Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositCurrentOct 22, 2026