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).

中文翻译：

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一些 Wronskian Hermite 多项式的不可约性

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，这些多项式因子是 x 乘以余数多项式的幂。我们证明余数多项式对于 m <= 2 的分区 (n, m) 和 (n, n) 当 n + 1 是平方时是不可约的。我们的主要工具是我们为所有分区证明的两个定理。第一个结果给出了余数多项式的牛顿多边形边缘斜率的陡峭上限。第二个结果是 Wronskian Hermite 多项式的 Schur 型同余。我们还解释了不可约性如何决定 Wronskian Hermite 多项式的实零点数，并证明 Veselov 对 (n, k, k-1, ..., 1) 形式的分区的猜想。