AI 中文总结
本文重新推导行列式$\boldsymbol{\nabla}_t$的计算方法并将其应用于雅可比多项式，得到其$\boldsymbol{\nabla}_t$及相关Hasse-Witt不变量的显式公式，所用技术具有广泛适用性。
AI 中文摘要
在文献[Feit]中，Feit利用广义拉盖尔多项式（GLP）证明了群$\tilde{A}_5$和$\tilde{A}_7$作为$\boldsymbol{Q}$上的伽罗瓦群存在。Hajir在文献[Hajir]中将该结果推广，证明当$n \not\bmod 8$时，$\tilde{A}_n$是$\boldsymbol{Q}$上的伽罗瓦群。两个证明的关键要素是明确确定GLP根域迹形式（对角化后）的Hasse-Witt不变量，这依赖于对某个行列式$\boldsymbol{\nabla}_t$的计算。文献[Feit]和[Hajir]中使用的$\boldsymbol{\nabla}_t$显式公式是利用GLP特有的性质推导的，该性质无法推广到其他多项式。本文重新考察Feit对$\boldsymbol{\nabla}_t$的原始计算，并将其置于汉克尔行列式的框架中；我们利用标准组合论证给出$\boldsymbol{\nabla}_t$的另一种推导，随后将这些结果应用于雅可比多项式——这是包含GLP作为特例的双参数正交多项式族。我们计算了雅可比多项式的$\boldsymbol{\nabla}_t$显式公式及相关的Hasse-Witt不变量。本文所用技术并非仅适用于雅可比多项式，具有广泛适用性。
英文摘要
In \cite{feit}, Feit used the Generalized Laguerre Polynomials (GLP) to prove that the groups $\widetilde{A}_{5}$ and $\widetilde{A}_{7}$ occur as Galois groups over $\Q$. Hajir, in \cite{hajir} extended these results to prove that $\widetilde{A}_{n}$ is Galois over $\Q$ whenever $n \equiv 1 \pmod{8}$. A key ingredient of both proofs is the explicit determination of the Hasse-Witt invariant of (the diagonalization of) the trace form of the root fields of the GLP, which relies on the calculation of a certain determinant, $Δ_t$. The explicit formula for $Δ_t$ used in \cite{feit} and \cite{hajir} was derived using properties specific to the GLP which do not generalize to other polynomials. In this paper we revisit Feit's original calculation of $Δ_t$ and situate it in the context of Hankel determinants. We give an alternate derivation of $Δ_t$ using standard combinatorial arguments and then apply these results to the Jacobi polynomials, a two-parameter family of orthogonal polynomials encompassing the GLP as a special case. We compute an explicit formula for the $Δ_t$ of the Jacobi polynomials as well as the associated Hasse-Witt invariant. The techniques used in this paper are not specific to the Jacobi polynomials and are widely applicable.
Comments16 pages
