Integral points on generic fibers

TL;DR

This paper investigates the conditions under which certain algebraic curves have infinitely many integral points and provides bounds on the number of such points for specific polynomial families.

Contribution

It characterizes when irreducible curves defined by rational polynomials have infinite integral points and identifies automorphisms transforming these polynomials into simpler forms.

Findings

Existence of automorphisms transforming P(x,y) to x or x^2 - dy^2.

Sharp bounds for the number of integral points with bounded coordinates.

Identification of conditions for infinite integral solutions on these curves.

Abstract

Let P(x,y) be a rational polynomial and k in Q be a generic value. If the curve (P(x,y)=k) is irreducible and admits an infinite number of points whose coordinates are integers then there exist algebraic automorphisms that send P(x,y) to the polynomial x or to x^2-dy^2. Moreover for such curves (and others) we give a sharp bound for the number of integral points (x,y) with x and y bounded.