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The L’vov–Kaplansky conjecture in degrees three and four over arbitrary fields

A mix of human-written and AI-generated textHuman understanding: some partsmath.RA — Rings and Algebras

Contributed by Qiyue Tang ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

We prove that for every field (K) and every positive integer (n), the image of each multilinear noncommutative polynomial of degree three or four on (M_n(K)) is a (K)-linear subspace.

Provenance statement

GPT-6-Pro was used across nine research sessions on the L’vov–Kaplansky conjecture, with a combined recorded duration of 55.5 hours. The proof for degrees three and four over arbitrary fields was completed during the eighth, nine-hour session. Codex was used to review the proofs and polish the manuscript. A Rethlas-OSV review is planned, and its results will be reported in a later version.

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Version history

  1. v1 · 59 pagesSubmitted by Qiyue TangInitial depositCurrentNov 07, 2026