On parametric and generic polynomials with one parameter

TL;DR

This paper investigates the relationship between one-parameter parametric polynomials and generic polynomials over fields, providing criteria for genericity, classifying such polynomials in characteristic zero, and constructing specific families of curves with rational points.

Contribution

It establishes that certain one-parameter parametric polynomials over complex fields are indeed generic, and offers a complete classification of one-parameter generic polynomials over characteristic zero fields.

Findings

Being L-parametric over a specific field implies the polynomial is generic.

Complete classification of one-parameter generic polynomials in characteristic zero.

Construction of affine curves with rational points but no rational generic point under BSD conjecture.

Abstract

Given fields $k \subseteq L$, our results concern one parameter $L$-parametric polynomials over $k$, and their relation to generic polynomials. The former are polynomials $P(T,Y) \in k[T][Y]$ 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 $L \supseteq k$. We show, for example, that being $L$-parametric with $L$ taken to be the single field $\mathbb{C}((V))(U)$ is in fact sufficient for a polynomial $P(T, Y) \in \mathbb{C}[T][Y]$ 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.