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A Proof of the Common Root Conjecture for Legendre Polynomials
Abstract
We prove Stieltjes' common root conjecture: Legendre polynomials of distinct degrees have no common nonzero root. We construct an auxiliary polynomial and show that, for the relevant power of two q≥2, the edge of slope 1/q in its $2$-adic Newton polygon has horizontal length less than 3q. If a common nonzero root existed, Newton polygon theory and the Legendre differential equation would force the same edge to have horizontal length at least 3q, yielding a contradiction.
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BibTeX
@misc{deng2026proof,
title = {{A Proof of the Common Root Conjecture for Legendre Polynomials}},
author = {Zikang Deng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22330},
eprint = {2609.22330},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22330},
}
RIS
TY - UNPB
TI - A Proof of the Common Root Conjecture for Legendre Polynomials
AU - Zikang Deng
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.22330
UR - https://arxiv.org/abs/2609.22330
AB - We prove Stieltjes' common root conjecture: Legendre polynomials of distinct degrees have no common nonzero root. We construct an auxiliary polynomial and show that, for the relevant power of two q≥2, the edge of slope 1/q in its $2$-adic Newton polygon has horizontal length less than 3q. If a common nonzero root existed, Newton polygon theory and the Legendre differential equation would force the same edge to have horizontal length at least 3q, yielding a contradiction.
ER -