Algebraic properties of a family of Jacobi polynomials
type: Item
instance of: scholarly article
title: Algebraic properties of a family of Jacobi polynomials
zbMATH Open document ID: 1239.11112
author: John Cullinan
author: Farshid Hajir
author: Elizabeth A. Sell
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2009-10-26T00:00:00Z
full work available at URL: https://eudml.org/doc/10878
review text: 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, 98…
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 5620670
zbMATH Keywords: Jacobi polynomials
zbMATH Keywords: irreducibility
zbMATH Keywords: Galois groups
zbMATH Keywords: Riemann-Hurwitz formula
reviewed by: Andreas Bender
MaRDI profile type: Publication
OpenAlex ID: W2312883437
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DOI: 10.5802/JTNB.659
description: scientific article; zbMATH DE number 5620670