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A Zariski-Open Family of Proper Septic Polynomials with Full Traceless Images

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

Let KK be an algebraically closed field of characteristic zero. We identify a 762762-dimensional space of proper multilinear degree-seven polynomials in the third power of the commutator ideal, with zero formal cyclic trace and annihilated by every repeated-input identification of type (3,3,1)(3,3,1). A nonempty Zariski-open subset of this space consists of polynomials pp whose sets of individual matrix values satisfy p(Mn(K)7)=sln(K)p(M_n(K)^7)=\mathfrak{sl}_n(K) for every n≥3n\geq3. The ambient proper and formal trace-zero spaces have dimensions 12601260 and 10491049, respectively. An explicit polynomial certifies nonemptiness of this coefficient-open family. The proof combines a symmetric-group decomposition, exact finite determinant witnesses, an additive-size argument and a separate 3×33\times3 matrix argument.

Provenance statement

Y. T. Miah selected the problem and directed the investigation. OpenAI's GPT-6 Pro contributed substantially to developing the mathematical arguments, computational verification, source analysis and manuscript preparation. The author reviewed the results and approved the final version. The work was first released on Zenodo on 8 November 2026 (DOI: 10.5281/zenodo.24060323). Its complete source, numerical data, five verification programs and finite certificates are available at https://github.com/oddcurios235/lvov-proper-septic-images.

Tools used

OpenAI
ChatGPTVersion 6 Pro

References

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  3. Kanel-Belov, Alexey; Malev, Sergey; Rowen, Louis. The images of multilinear polynomials evaluated on 3×33\times3 matrices. Proc. Amer. Math. Soc., vol. 144, no. 1, pp. 7–19. 2016. doi:10.1090/proc/12478 (https://doi.org/10.1090/proc/12478)link
  4. Miah, Y. T. The L'vov–Kaplansky Conjecture for Standard Polynomials. 2026. doi:10.5281/zenodo.24031055 (https://doi.org/10.5281/zenodo.24031055). Hexagon preprint 2611.00184v1 (https://hexagonmath.org/2611.00184v1)link
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Version history

  1. v1 · 12 pagesSubmitted by Y. T. MiahInitial depositCurrentNov 08, 2026