On the Genus of Generalized Laguerre Polynomials

belong to one of the three family of orthogonal polynomials, the other two being Jacobi and Legendre. In addition to their important roles in mathematical analysis, these polynomials also feature prominently in algebra and number theory. Schur ([7], [8]) pioneered the study of Galois properties of specializations of these orthogonal polynomials, and Feit [1] used them to solve the inverse Galois problem over Q for certain double covers of the alternating group An; see ([3], [4]) for other related results. Recently Hajir and Wong [5] proved that for n ≥ 5 and for any number field K, the Galois group of L n (x) over K is Sn for all but finitely many α ∈ K. A key ingredient of the proofs is to compute the genus of the function fields in the splitting field of L (t) n (x) over the function field Q(t). As a by-product of this argument, we showed that for n ≥ 5, L n (x) defines an absolutely irreducible plane curve Ln of geometric genus > 1. In light of the importance of L n (x) in algebra and other areas of mathematics, in this paper we determine the exact genus of these curves. Theorem. For n ≥ 1, the equation L n (x) = 0 defines an absolutely irreducible plane curve of geometric genus [(n− 1)/2][(n− 2)/2] = [(−1 + n/2)].