On parametric and generic polynomials with one parameter

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.