Algebraic functions with infinitely many values in a number field

TL;DR

This paper characterizes algebraic curves over algebraic closures of rationals where infinitely many points in a number field yield polynomial roots within that same field, revealing special algebraic properties.

Contribution

It provides a classification of algebraic curves with infinitely many points over a number field where the polynomial roots are contained in that field.

Findings

Identification of conditions for algebraic curves with infinitely many such points

Characterization of the algebraic structure of these curves

Insights into the distribution of roots over number fields

Abstract

We describe algebraic curves $ X : F(x, y) = 0 $ defined over $\overline{\mathbb{Q}}$ that satisfy the following property: there exist a number field $k$ and an infinite set $S \subset k$ such that, for every $y \in S$, the roots of the polynomial $F(x, y)$ belong to $k$.