论文标题
某些Wronskian Hermite多项式的不可约性
The irreducibility of some Wronskian Hermite polynomials
论文作者
论文摘要
我们研究了由分区标记的Wronskian Hermite多项式的不可约性。众所周知，这些多项式因素作为X倍量其余多项式的功能。我们表明，当n + 1是正方形时，其余多项式对于具有m <= 2的分区（n，m）是不可修复的。 我们的主要工具是我们为所有分区证明的两个定理。第一个结果为剩余多项式的牛顿多边形边缘的斜率提供了尖锐的上限。第二个结果是Schur-Type的一致性。 我们还解释了如何确定Wronskian Hermite多项式的真实零的数量，并证明了Veselov对形式分区的猜想（N，K，K-1，...，1）。
We study the irreducibility of Wronskian Hermite polynomials labelled by partitions. It is known that these polynomials factor as a power of x times a remainder polynomial. We show that the remainder polynomial is irreducible for the partitions (n, m) with m <= 2, and (n, n) when n + 1 is a square. Our main tools are two theorems that we prove for all partitions. The first result gives a sharp upper bound for the slope of the edges of the Newton polygon for the remainder polynomial. The second result is a Schur-type congruence for Wronskian Hermite polynomials. We also explain how irreducibility determines the number of real zeros of Wronskian Hermite polynomials, and prove Veselov's conjecture for partitions of the form (n, k, k-1, ..., 1).
