On Galois groups of a one-parameter orthogonal family of polynomials
Volume 183 / 2018
Abstract
For a fixed integer $t \gt 1$, we show that if $t$ is not equal to $2$, a square $\ge 4$, or three times a square, then the discriminant of the generalized Laguerre polynomial $L_{n}^{(s/t)}(x)$ is a nonzero square for at most finitely many pairs $(n,s)$. Otherwise, the discriminant of $L_{n}^{(s/t)}(x)$ is a nonzero square if and only if $(n,s)$ belongs to one of seven explicitly describable infinite sets or to an additional finite set. This extends the results obtained for $t=1$ by P. Banerjee, M. Filaseta, C. Finch and J. Leidy. As a consequence, if $\alpha$ is a fixed rational number not equal to $1$, $3$, $5$, or a negative integer, then for all but finitely many $n$, $L_{n}^{(\alpha)}(x)$ has Galois group $S_{n}$, thereby refining a previous result of M. Filaseta – T. Y. Lam and F. Hajir. As an illustration, we give for $t=2$ infinitely many integer specializations $(n,s(n))$ such that $L_{n}^{(s(n)/2)}(x)$ has Galois group $A_{n}$. For $n \le 5$, the set of rational numbers $\alpha$ for which the discriminant of $L_{n}^{(\alpha)}(x)$ is a nonzero square is explicitly computed by solving certain generalized Pell-like equations.
