卡萨-阿尔韦罗猜想的下降集障碍
A descent-set obstruction for the Casas-Alvero conjecture
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中文总结 AI 辅助
本文针对卡萨-阿尔韦罗猜想，通过归约得到其下降集障碍，证明反例需至少 3 个回收根，给出特征 p 下坏素数的判据，为已知情形的相关命题提供简短证明。
中文摘要 AI 辅助
卡萨-阿尔韦罗猜想断言：特征零域上次数为 d 的首一多项式 f，若与 f',…,f^{(d-1)} 中每一个都共享一个非平凡因子，则 f 必为某个线性多项式的 d 次幂。本文完成了两次归约：将一个根平移至原点后，仅当系数 a_{d-i} 非零时，第 i 个索引对应的条件才非空，因此卡萨-阿尔韦罗轨迹可通过中心化范式的支撑集分层；若进一步指定每个存活条件由哪个根实现，则该系统将成为单位下三角矩阵，且完全消去 f 的系数。当仅需一个根时，剩余部分为单个整数，该整数即 MacMahon 行列式：集合 {1,…,d} 中下降集为 f 的支撑集的排列数。由于每个子集都是下降集，不存在反例的回收根数量少于 3 个；在特征 p（p 为上述计数之一的素因子，但无伴随整数 z_i 为坏素数）时，d 次的最大计数为欧拉锯齿数 A_d，故不规则素数 691 对 d=11 而言是坏素数。该分层还为所有已知情形所依赖的两个特征 p 命题提供了简短证明；任意特征下均单独确定两个元素的支撑集，由此可得反例的中心化范式至少有 4 项，且存在一种无需 Gröbner 基的坏素数判据。
英文摘要
The Casas--Alvero conjecture asserts that a monic polynomial $f$ of degree $d$ over a field of characteristic zero sharing a non-constant factor with each of $f',\dots,f^{(d-1)}$ is the $d$-th power of a linear polynomial. Two reductions are carried out. Once a root is translated to the origin, the condition at index $i$ is vacuous unless the coefficient $a_{d-i}$ is nonzero, so the Casas--Alvero locus stratifies by the support of the centred normal form; prescribing in addition which root realises each surviving condition makes the system unit lower triangular and eliminates the coefficients of $f$ entirely. When one root suffices, what remains is a single integer, and that integer is MacMahon's determinant: the number of permutations of $\{1,\dots,d\}$ with descent set the support of $f$. Since every subset is a descent set, no counterexample has fewer than three recycled roots; in characteristic $p$ a prime dividing one of these counts but none of the accompanying integers $z_{i}$ is a bad prime, the largest count in degree $d$ being the Euler zigzag number $A_{d}$, so that the irregular prime $691$ is bad for $d=11$. The stratification also yields short proofs of the two characteristic-$p$ propositions on which all known cases rest; supports of two elements are settled separately in every characteristic, whence a counterexample has at least four terms in centred normal form and a further Gröbner-free criterion for bad primes.
