Level curves of rational functions and unimodular points on rational curves

TL;DR

This paper generalizes previous results on zeros of polynomial powers by analyzing intersections of level curves of complex functions using complex analysis and algebraic geometry.

Contribution

It extends earlier work on polynomial zeros to a broader context involving level curves of rational functions and unimodular points on rational curves.

Findings

Improved bounds on common zeros of shifted polynomial powers.

Generalized the problem to intersections of level curves of complex functions.

Applied classical complex analysis and algebraic geometry tools.

Abstract

We obtain an improvement and broad generalisation of a result of N. Ailon and Z. Rudnick (2004) on common zeros of shifted powers of polynomials. Our approach is based on reducing this question to a more general question of counting intersections of level curves of complex functions. We treat this question via classical tools of complex analysis and algebraic geometry.