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

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.

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.

DOI : 10.5802/jtnb.659
Mots clés : Orthogonal polynomials, Jacobi polynomial, Rational point, Riemann-Hurwitz formula, Specialization
Cullinan, John 1 ; Hajir, Farshid 2 ; Sell, Elizabeth 3

1 Department of Mathematics Bard College Annandale-On-Hudson, NY 12504
2 Department of Mathematics University of Massachusetts Amherst MA 01003
3 Department of Mathematics Millersville University P.O. Box 1002 Millersville, PA 17551
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Cullinan, John; Hajir, Farshid; Sell, Elizabeth. 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. http://www.numdam.org/articles/10.5802/jtnb.659/

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