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Arithmetic of Jacobi polynomials

A mix of human-written and AI-generated textHuman understanding: some partsmath.GM — General Mathematics

Contributed by Khai Hoan Nguyen Dang ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

We study the arithmetic of Jacobi polynomials. As applications, we prove the Mahlburg--Ono conjecture on the irreducibility and Galois groups of the four Kaneko--Zagier families, and the Cullinan--Hajir conjecture on the Galois groups of Legendre polynomials. Other applications concern common zeros of Jacobi polynomials, harmonic functions on cones, and hyperelliptic Jacobians.

Provenance statement

This work was developed by the author with assistance from ChatGPT (OpenAI). The author developed the overall approach and methods, building on the prior work cited in the manuscript. During the research and preparation of the manuscript, ChatGPT was used to discuss and revise mathematical arguments, improve the exposition and search for relevant literature. It also assisted with all computations presented in the appendices.

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Version history

  1. v1Submitted by Khai Hoan Nguyen DangInitial depositCurrentOct 22, 2026