Diophantine equations in separated variables and polynomial power sums

Area of Science:

• Number Theory
• Algebraic Geometry

Background:

• Diophantine equations are fundamental in number theory, seeking integer or rational solutions.
• Power sum polynomials are a specific class of polynomials with applications in various mathematical fields.

Purpose of the Study:

• To investigate the nature and frequency of rational solutions for Diophantine equations where both sides are power sums.
• To determine conditions under which infinitely many rational solutions with bounded denominators can exist.

Main Methods:

• The study employs the powerful finiteness criterion developed by Bilu and Tichy.
• Analysis of Diophantine equations of the form f(x) = g(y), where f and g are power sums.

Main Results:

• The research demonstrates that, under specific assumptions, infinitely many rational solutions (x, y) with bounded denominators are generally not possible.
• The study identifies that such infinite solutions are restricted to trivial cases.

Conclusions:

• The Bilu-Tichy criterion effectively limits the existence of abundant rational solutions for this class of Diophantine equations.
• The findings contribute to understanding the distribution and limitations of rational points on curves defined by power sum polynomials.