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The L’vov–Kaplansky Conjecture for Standard Polynomials

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

Contributed by Y. T. Miah ↗

Version 1 / Nov 07, 2026 / CC BY 4.0

Abstract

The L'vov–Kaplansky conjecture asks whether the image of a multilinear polynomial on a full matrix algebra is a vector subspace. For the unnormalized standard polynomial sds_d over an algebraically closed field KK of characteristic zero, the sets of individual matrix values are determined in every degree and order. The image of s2r−1s_{2r-1} is zero when n<rn<r and all of Mn(K)M_n(K) when n≥rn\ge r; the image of s2rs_{2r} is zero when n≤rn\le r and sln(K)\mathfrak{sl}_n(K) when n≥r+1n\ge r+1. The odd-degree proof uses explicit linear slices and Jordan forms. The even-degree argument combines a uniform nilpotent construction and an additive transfer for admissible cores with generic-nullity results of Brassil–Reichstein and Lin–Tang–Zhao, extending a method of Qiyue Tang to obtain actual polynomial values.

Provenance statement

Y. T. Miah selected the problem and directed the investigation. OpenAI's GPT-6 Pro contributed substantially to developing the odd- and even-degree mathematical arguments, performing exact computations, investigating the literature, and preparing the manuscript. The author reviewed the results and approved the final version. The work was first released on Zenodo on 7 November 2026 (DOI: 10.5281/zenodo.24031055). The complete source, verification programs and finite certificates are available at https://github.com/oddcurios235/lvov-kaplansky-standard-polynomials.

Tools used

OpenAI
ChatGPTVersion 6 Pro

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Version history

  1. v1 · 30 pagesSubmitted by Y. T. MiahInitial depositCurrentNov 07, 2026