Algebraic properties of a family of Jacobi polynomials

John Cullinan; Farshid Hajir; Elizabeth Sell

doi:10.5802/jtnb.659

John Cullinan ; Farshid Hajir ; Elizabeth Sell

Journal de Théorie des Nombres de Bordeaux, Tome 21 (2009) no. 1, pp. 97-108.

Résumé
Abstract

La famille des polynômes à un seul paramètre $J_{n} (x, y) = \sum_{j = 0}^{n} (\binom{y + j}{j}) x^{j}$ est une sous-famille de la famille (à deux paramètres) des polynômes de Jacobi. On montre que pour chaque $n \geq 6$ , quand on spécialise en $y_{0} \in ℚ$ , le polynôme $J_{n} (x, y_{0})$ est irréductible sur $ℚ$ , sauf pour un nombre fini des valeurs $y_{0} \in ℚ$ . Si $n$ est impair, sauf pour un nombre fini des valeurs $y_{0} \in ℚ$ , le groupe de Galois de $J_{n} (x, y_{0})$ est $S_{n}$ ; si $n$ est pair, l’ensemble exceptionnel est mince.

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 $n \geq 6$ , the polynomial $J_{n} (x, y_{0})$ is irreducible over $ℚ$ for all but finitely many $y_{0} \in ℚ$ . 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.

Publié le : 2009-08-24

MR 2537705 | Zbl pre05620670

DOI : https://doi.org/10.5802/jtnb.659
Mots clés : Orthogonal polynomials, Jacobi polynomial, Rational point, Riemann-Hurwitz formula, Specialization

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     author = {John Cullinan and Farshid Hajir and Elizabeth Sell},
     title = {Algebraic properties of a family of Jacobi polynomials},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {97--108},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {1},
     year = {2009},
     doi = {10.5802/jtnb.659},
     zbl = {pre05620670},
     mrnumber = {2537705},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2009__21_1_97_0/}
}

John Cullinan; Farshid Hajir; Elizabeth Sell. Algebraic properties of a family of Jacobi polynomials. Journal de Théorie des Nombres de Bordeaux, Tome 21 (2009) no. 1, pp. 97-108. doi : 10.5802/jtnb.659. https://jtnb.centre-mersenne.org/item/JTNB_2009__21_1_97_0/

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