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John Cullinan; Farshid Hajir; Elizabeth Sell
Algebraic properties of a family of Jacobi polynomials
Journal de théorie des nombres de Bordeaux, 21 no. 1 (2009), p. 97-108, doi: 10.5802/jtnb.659
Article PDF | Reviews MR 2537705 | Zbl pre05620670
Keywords: Orthogonal polynomials, Jacobi polynomial, Rational point, Riemann-Hurwitz formula, Specialization

Résumé - Abstract

The one-parameter family of polynomials $J_{n}(x,y) = \sum _{j=0}^{n} \binom{y+j}{j}x^{j}$ is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each $n \ge 6$, the polynomial $J_{n}(x,y_{0})$ is irreducible over $\mathbb{Q}$ for all but finitely many $y_{0} \in \mathbb{Q}$. If $n$ is odd, then with the exception of a finite set of $y_{0}$, the Galois group of $J_{n}(x,y_{0})$ is $S_{n}$; if $n$ is even, then the exceptional set is thin.

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