2021/02/15 by Dèbes, Pierre, König, Joachim, Legrand, François +1
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2102.07465
Given fields k ⊆ L, our results concern one parameter L-parametric polynomials over k, and their relation to generic polynomials. The former are polynomials P(T,Y) ∈ k[T][Y] of group G which parametrize all Galois extensions of L of group G via specialization of T in L, and the latter are those which are L-parametric for every field L ⊇ k. We show, for example, that being L-parametric with L taken to be the single field ℂ((V))(U) is in fact sufficient for a polynomial P(T, Y) ∈ ℂ[T][Y] to be generic. As a corollary, we obtain a complete list of one parameter generic polynomials over a given field of characteristic 0, complementing the classical literature on the topic. Our approach also applies to an old problem of Schinzel: subject to the Birch and Swinnerton-Dyer conjecture, we provide one parameter families of affine curves over number fields, all with a rational point, but with no rational generic point.