HEXAGONbeta

The L'vov–Kaplansky conjecture for polynomials of degree four

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

Contributed by Qiyue Tang ↗

Version 1 / Nov 05, 2026 / CC BY 4.0

Abstract

We prove the L'vov–Kaplansky conjecture for multilinear polynomials of degree four over algebraically closed fields of characteristic zero.

Provenance statement

The mathematical results in this manuscript were developed through two research conversations with GPT-6-Pro, each spanning a 3.5-hour research window, for a total of seven hours. Codex was used to review the arguments, organize the manuscript, and polish its exposition. The manuscript also underwent review using Rethlas-OSV.

References

  1. K. Shoda, Einige Sätze über Matrizen , Japanese J. Math. 13 (1936), 361--365, doi:10.4099/jjm1924.13.0_361 (https://doi.org/10.4099/jjm1924.13.0_361).DOI
  2. A. A. Albert and B. Muckenhoupt, On matrices of trace zeros , Michigan Math. J. 4 (1957), 1--3.
  3. S. A. Amitsur and J. Levitzki, Minimal identities for algebras , Proc. Amer. Math. Soc. 1 (1950), 449--463.
  4. E. Formanek, Central polynomials for matrix rings , J. Algebra 23 (1972), 129--132.
  5. M. Brešar and I. Klep, Values of noncommutative polynomials, Lie skew-ideals and tracial Nullstellensätze , Math. Res. Lett. 16 (2009), 605--626; arXiv:0810.1774 (https://arxiv.org/abs/0810.1774).arXiv
  6. Z. Mesyan, Polynomials of small degree evaluated on matrices , Linear Multilinear Algebra 61 (2013), 1487--1495; arXiv:1212.1925 (https://arxiv.org/abs/1212.1925).arXiv
  7. D. Buzinski and R. Winstanley, On multilinear polynomials in four variables evaluated on matrices , Linear Algebra Appl. 439 (2013), 2712--2719; arXiv:1511.06331 (https://arxiv.org/abs/1511.06331).arXiv
  8. A. Kanel-Belov, S. Malev, and L. Rowen, The images of multilinear polynomials evaluated on 3×33\times3 matrices , Proc. Amer. Math. Soc. 144 (2016), 7--19; arXiv:1306.4389v3 (https://arxiv.org/abs/1306.4389).arXiv
  9. A. Kanel-Belov, S. Malev, L. Rowen, and R. Yavich, Evaluations of noncommutative polynomials on algebras: methods and problems, and the L'vov--Kaplansky conjecture , arXiv:1909.07785v4 (https://arxiv.org/abs/1909.07785).arXiv
  10. A. Kanel-Belov, S. Malev, and L. Rowen, The images of non-commutative polynomials evaluated on 2×22\times2 matrices , Proc. Amer. Math. Soc. 140 (2012), 465--478.
Show all 15 referencesShow fewer references
  1. K. Dykema and I. Klep, Instances of the Kaplansky--Lvov multilinear conjecture for polynomials of degree three , Linear Algebra Appl. 508 (2016), 272--288.
  2. D. Vitas, The L'vov--Kaplansky conjecture for polynomials of degree three , Linear Algebra Appl. 733 (2026), 205--232, doi:10.1016/j.laa.2025.12.016 (https://doi.org/10.1016/j.laa.2025.12.016); arXiv:2310.15600v2 (https://arxiv.org/abs/2310.15600).arXivDOI
  3. P. S. Fagundes, T. C. de Mello, and P. H. da Silva dos Santos, The Mesyan conjecture: a restatement and a correction , arXiv:2111.13698v1 (https://arxiv.org/abs/2111.13698).arXiv
  4. M. de Bondt, Mathieu subspaces of codimension less than nn of Mat⁡n(K)\operatorname{Mat}_n(K) , arXiv:1310.7843v2 (https://arxiv.org/abs/1310.7843).arXiv
  5. P. A. Fillmore, On similarity and the diagonal of a matrix , Amer. Math. Monthly 76 (1969), 167--169.

Version history

  1. v1 · 28 pagesSubmitted by Qiyue TangInitial depositCurrentNov 05, 2026