HEXAGONbeta

Actual images and local fibers in the second commutator layer

Primarily human-written textHuman understanding: all partsmath.RA — Rings and Algebras

Contributed by Shengyong Pan ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

Let KK be an algebraically closed field of characteristic zero and let JJ be the commutator ideal of the free associative algebra. We prove that, for every proper multilinear polynomial f∈J2∖J3f\in J^2\setminus J^3 and every n≥3n\ge3, the set of values of ff on Mn(K)M_n(K) is sln(K)\mathfrak{sl}_n(K) if $\tr f\equiv0$, and Mn(K)M_n(K) otherwise. In the first case the evaluation map is smooth at an input with nonzero rank-one square-zero value. In the second case it is flat at such an input, with a local complete-intersection fiber of the expected dimension. The theorem allows arbitrary proper multilinear corrections of the same degree in J3J^3. We use restrictions to two active variables with commuting passive inputs and detect the first nonzero trace term on the exact fiber of the traceless projection.

References

  1. P. A. Fillmore, On similarity and the diagonal of a matrix , Amer. Math. Monthly 76 (1969), 167--169. https://doi.org/10.1080/00029890.1969.12000162.DOI
  2. 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 , SIGMA 16 (2020), 071. https://arxiv.org/abs/1909.07785.link
  3. I. Gonzales Gargate and T. Castilho de Mello, Images of multilinear polynomials on n×nn\times n upper triangular matrices over infinite fields , Israel J. Math. 252 (2022), 337--354. https://arxiv.org/abs/2106.12726.link
  4. A. S. Amitsur and J. Levitzki, Minimal identities for algebras , Proc. Amer. Math. Soc. 1 (1950), 449--463.
  5. S. Pan, The L'vov--Kaplansky conjecture for multilinear polynomials of degree four , manuscript, 18 September 2026.
  6. S. Pan, The L'vov--Kaplansky conjecture for multilinear polynomials of degrees four to six , preprint (2026), Hexagon 2611.00173v1 (https://hexagonmath.org/2611.00173v1).hexagon
  7. S. Pan, Actual images and local fibers of proper matrix polynomials , research manuscript, 6 November 2026.
  8. S. Pan, Actual matrix values and the strengthened Mesyan conjecture: general constructions and higher-degree applications , research manuscript, 9 November 2026.
  9. Q. Tang, The L'vov--Kaplansky conjecture for polynomials of degree four , preprint (2026), Hexagon 2611.00119v1 (https://hexagonmath.org/2611.00119v1).hexagon
  10. I. Glazer and Y. I. Hendel, On singularity properties of word maps and applications to probabilistic Waring type problems , arXiv:1912.12556. https://arxiv.org/abs/1912.12556.arXiv
Show all 16 referencesShow fewer references
  1. The Stacks Project, Lemma 29.35.10. https://stacks.math.columbia.edu/tag/01V4.link
  2. The Stacks Project, Lemma 10.128.1. https://stacks.math.columbia.edu/tag/00R4.link
  3. The Stacks Project, Section 37.15, Theorem 37.15.1. https://stacks.math.columbia.edu/tag/0398.link
  4. The Stacks Project, Lemma 29.26.10. https://stacks.math.columbia.edu/tag/01UA.link
  5. 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, Theorem 2. https://doi.org/10.1090/S0002-9939-2011-10963-8. Final author manuscript: https://arxiv.org/pdf/1005.0191v3, p. 3.DOI
  6. 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), no. 1, 7--19. https://doi.org/10.1090/proc/12478. Author version arXiv:1306.4389v3, Lemmas 1--2, p. 2: https://arxiv.org/pdf/1306.4389v3.arXivDOI

Version history

  1. v1 · 49 pagesSubmitted by Shengyong PanInitial depositCurrentNov 09, 2026