Algebraic properties of a family of Jacobi polynomials
The topic of this article is the family of polynomials \[ J_n = \sum_{j=0}^{n}{y+j \choose j}x^j \] with the following main result. \textbf{Theorem} For any integer \(n\geq 6\), the polynomials \(J_n(x, y_0)\) are irreducible over \(\mathbb{Q}\) for all but finitely many \(y_0\in \mathbb{Q}\). If \(n\) is odd, then the Galois group of \(J_n(x, y_0)\) is equal to \(S_n\) for all but finitely many \(y_0\in \mathbb{Q}\). If \(n\) is even, there is a thin set of \(y_0\) for which the Galois group is \(A_n\). One of the tools used in the proof involves Faltings' theorem, so the proof is noneffective. The authors outline the strategy of their original proof involving a comparison of the \(p\)-adic Newton polygons of the target polynomials and of the truncated exponential series. This approach is effective but is described by the authors as more intricate and was therefore replaced with the noneffective argument. By a simple transformation, the authors replace the \(J_n\) with the family \[ P_n(x,y)=\sum_{j=0}^{n}{y\choose j}x^{n-j}, \] which has equivalent algebraic properties. The first step consists of showing that the Galois group of \(P_n(x,y)\) over \(\mathbb{Q}(y)\) is \(S_n\). Irreducibility is deduced from a straightforward application of Eisenstein's criterion, the type of Galois group follows from considering the explicit discriminant polynomial of \(P_n(x,y)\) which allows to determine enough inertia groups contained in the Galois group. With the algebraic closure of \(P_n(x,y)=0\) in the projective plane denoted by \(X_1\), the genus of a normalization of \(X_1\) is determined by application of the Riemann-Hurwitz formula to the covering of \(\mathbb{P}^1\) by \(X_1\). The proof of the main theorem involves proving that the fixed fields of maximal subgroups of \(A_n\) all have genus larger than 1. The basic idea is a technique given in section 4.5 [\textit{P. Müller}, ``Finiteness results for Hilbert's irreducibility theorem'', Ann. Inst. Fourier 52, No. 4, 983--1015 (2002; Zbl 1014.12002)], the calculations use a known classification of such maximal subgroups and consist of explicit estimates of the respective genera.
scientific article; zbMATH DE number 5620670
scientific article