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。AI 中文总结
本文证明Stieltjes提出的Legendre多项式不可约性猜想，通过结式与导数恒等式比较界，排除n≥100000的因式分解，其余由已有结果覆盖。
AI 中文摘要
我们证明了Stieltjes在1890年致Hermite的信中提出的Legendre多项式不可约性猜想。假设存在非平凡因式分解，我们取属于不同因子的根之间的差，将这些差相乘，并包含首项系数，得到一个非零整数，即两个因子的结式。我们首先界定每个奇素数整除该整数的指数，得到其奇数部分的上界。然后利用由Legendre微分方程导出的根处导数恒等式，证明同一奇数部分超过另一个显式量。比较这两个界排除了原始次数$n \ge 100000$的所有因式分解。其余次数由Groth先前建立的有限范围结果覆盖。
英文摘要
We prove the irreducibility conjecture for Legendre polynomials proposed by Stieltjes: for every integer $j\geq 1$, both $P_{2j}(x)$ and $P_{2j+1}(x)/x$ are irreducible over the rational numbers. The proof proceeds by contradiction. Starting from a hypothetical nontrivial factorization, we remove the zero root from $P_n$ when present, multiply by a suitable constant, and express the resulting polynomial as a product of two integral polynomials $A(x^2)$ and $B(x^2)$. We then construct the resultant and its odd part, $R=\operatorname{Res}_t(A,B), \mathcal{R}=\frac{|R|}{2^{v_2(R)}}.$ Put $k=\operatorname{deg} A\leq\operatorname{deg} B$ and $m=\lfloor n/2\rfloor$. Using the classical auxiliary polynomial $U_n$ and the differential identity $(1-x^2)(P_n'U_n-P_nU_n')=1-P_n^2,$ together with orthogonality, divisibility properties of the coefficients, and a least-common-multiple estimate, we obtain an upper bound for $\mathcal{R}$. On the other hand, the Legendre differential equation gives a lower bound for the absolute value of the derivative of $AB$ at each root of $A$. We use Chebyshev polynomials and Hadamard's inequality to bound the product of the squared pairwise differences of these roots, and combine this with estimates for the leading coefficients and the power of $2$ in $R$ to obtain a lower bound for $\mathcal{R}$. These estimates yield $m\log4-\log\frac{4(m+1)^2}{\sqrt m} <\frac{\log\mathcal R}{k} <(m+\sqrt{2n-1})\log3 (n\ge64).$ For every $n\ge256$, the lower bound strictly exceeds the upper bound, giving a contradiction. Combining this with established irreducibility results and explicit integer comparisons for the remaining degrees proves irreducibility in every degree.
