Representation of Unity by Binary Forms

TL;DR

This paper establishes upper bounds on the number of integer solutions to certain binary form equations, using a combination of classical analysis, geometry of numbers, and linear forms in logarithms.

Contribution

It provides new upper bounds for solutions of |F(x,y)|=1 for irreducible binary forms with large discriminant, improving understanding of these Diophantine equations.

Findings

At most 11n-2 solutions for large discriminant

Sharper bounds under additional restrictions

Methods combine analysis, geometry of numbers, and linear forms in logarithms

Abstract

In this paper, it is shown that if F(x , y) is an irreducible binary form with integral coefficients and degree $n \geq 3$, then provided that the absolute value of the discriminant of F is large enough, the equation |F(x , y)| = 1 has at most 11n-2 solutions in integers x and y. We will also establish some sharper bounds when more restrictions are assumed. These upper bounds are derived by combining methods from classical analysis and geometry of numbers. The theory of linear forms in logarithms plays an essential role in studying the geometry of our Diophantine equations.