勒让德多项式公共根猜想的一个证明

中文摘要 AI 辅助

我们证明了斯蒂尔杰斯的公共根猜想：不同次数的勒让德多项式没有公共的非零根。我们构造了一个辅助多项式，并证明对于相关的2的幂$q\ge2$，其2-adic牛顿多边形中斜率为$1/q$的边的水平长度小于$3q$。若存在公共非零根，则牛顿多边形理论和勒让德微分方程将迫使同一条边的水平长度至少为$3q$，从而产生矛盾。

英文摘要

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 $q\ge2$, 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.

发表机构

• Beijing Normal University（北京师范大学）

机构由 AI 辅助整理，请以论文原文为准。