Algebraic properties of a family of Jacobi polynomials

Cullinan, John; Hajir, Farshid; Sell, Elizabeth

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

La famille des polynômes à un seul paramètre $J_n(x,y)={\sum }_{j=0}^n\left(\genfrac{}{}{0pt}{}{y+j}{j}\right)x^j$ est une sous-famille de la famille (à deux paramètres) des polynômes de Jacobi. On montre que pour chaque $n\ge 6$, quand on spécialise en $y_0\in \mathbb{Q}$, le polynôme $J_n(x,y_0)$ est irréductible sur $\mathbb{Q}$, sauf pour un nombre fini des valeurs $y_0\in \mathbb{Q}$. Si $n$ est impair, sauf pour un nombre fini des valeurs $y_0\in \mathbb{Q}$, 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 $J_n(x,y)={\sum }_{j=0}^n\left(\genfrac{}{}{0pt}{}{y+j}{j}\right)x^j$ 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.

Mots clés : Orthogonal polynomials, Jacobi polynomial, Rational point, Riemann-Hurwitz formula, Specialization

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