Algebraic properties of a family of Jacobi polynomials
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, p. 97-108

The one-parameter family of polynomials J n (x,y)= j=0 n y+j jx j is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each n6, the polynomial J n (x,y 0 ) is irreducible over for all but finitely many y 0 . 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.

La famille des polynômes à un seul paramètre J n (x,y)= j=0 n y+j jx j est une sous-famille de la famille (à deux paramètres) des polynômes de Jacobi. On montre que pour chaque n6, quand on spécialise en y 0 , le polynôme J n (x,y 0 ) est irréductible sur , sauf pour un nombre fini des valeurs y 0 . Si n est impair, sauf pour un nombre fini des valeurs y 0 , le groupe de Galois de J n (x,y 0 ) est S n  ; si n est pair, l’ensemble exceptionnel est mince.

DOI : https://doi.org/10.5802/jtnb.659
Keywords: Orthogonal polynomials, Jacobi polynomial, Rational point, Riemann-Hurwitz formula, Specialization
@article{JTNB_2009__21_1_97_0,
     author = {Cullinan, John and Hajir, Farshid and Sell, Elizabeth},
     title = {Algebraic properties of a family of Jacobi polynomials},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {1},
     year = {2009},
     pages = {97-108},
     doi = {10.5802/jtnb.659},
     mrnumber = {2537705},
     zbl = {pre05620670},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2009__21_1_97_0}
}
Cullinan, John; Hajir, Farshid; Sell, Elizabeth. Algebraic properties of a family of Jacobi polynomials. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, pp. 97-108. doi : 10.5802/jtnb.659. http://www.numdam.org/item/JTNB_2009__21_1_97_0/

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