A Chebyshev criterion for Abelian integrals

TL;DR

This paper introduces a new algebraic criterion to determine when Abelian integrals exhibit the Chebyshev property, simplifying existing proofs and enabling new results in the study of these integrals.

Contribution

The paper proposes a novel algebraic criterion for the Chebyshev property of Abelian integrals, streamlining proofs and extending applicability beyond standard methods.

Findings

The criterion simplifies checking the Chebyshev property.

Several known results are derived more succinctly.

New results not accessible by previous methods are obtained.

Abstract

We present a criterion that provides an easy sufficient condition in order that a collection of Abelian integrals has the Chebyshev property. This condition involves the functions in the integrand of the Abelian integrals and can be checked, in many cases, in a purely algebraic way. By using this criterion, several known results are obtained in a shorter way and some new results, which could not be tackled by the known standard methods, can also be deduced.