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Images of Multilinear Polynomials with a Standard-Polynomial Unit Specialization

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

Contributed by Y. T. Miah ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

For r=3,4,5r=3,4,5, consider an ordinary multilinear polynomial f(Y,X1,…,X2r)f(Y,X_1,\ldots,X_{2r}) whose specialization at Y=IY=I is a nonzero scalar multiple of the standard polynomial s2rs_{2r}. Over an algebraically closed field KK of characteristic zero, we prove that for every n≥r+1n\ge r+1 the set of its individual values on Mn(K)M_n(K) is sln(K)\mathfrak{sl}n(K) if its trace vanishes identically, and Mn(K)M_n(K) otherwise. This includes arbitrary proper perturbations of s7,s9,s11s_7,s_9,s{11}. The proof combines exact standard-polynomial core constructions, additive size transfer, and a mixed-frame compatibility lemma with a local lifting criterion of Pan.

Provenance statement

Y. T. Miah selected the problem and directed the investigation. OpenAI's GPT-6 Pro contributed substantially to the mathematical arguments, proof development, exact computations, literature review and manuscript drafting. The author reviewed and approved the final version. This paper was first released on Zenodo on 8 November 2026 (DOI: 10.5281/zenodo.24060041). The complete source, verification program and finite certificates are available at https://github.com/oddcurios235/standard-unit-images.

Tools used

OpenAI
ChatGPTVersion 6 Pro

References

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  4. Miah, Y. T. The L'vov–Kaplansky Conjecture for Standard Polynomials. 2026. 10.5281/zenodo.24031055. Hexagon preprint 2611.00184v1 (https://hexagonmath.org/2611.00184v1)hexagon
  5. Pan, Shengyong. The L'vov–Kaplansky conjecture for multilinear polynomials of degrees four to six. 2026. Hexagon preprint 2611.00173v1 (https://hexagonmath.org/2611.00173v1)hexagon
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Version history

  1. v1 · 9 pagesSubmitted by Y. T. MiahInitial depositCurrentNov 08, 2026