Principal Poincar\'e Pontryagin Function associated to some families of Morse real polynomials

TL;DR

This paper investigates conditions under which the Principal Poincaré Pontryagin Function, associated with families of Morse real polynomials, is an Abelian integral or an iterated integral, extending previous results to non-isodromic cases.

Contribution

It provides a sufficient monodromy condition ensuring the Principal Poincaré Pontryagin Function is an Abelian integral beyond generic cases, and extends known results to certain non-isodromic families.

Findings

Identifies monodromy conditions for Abelian integral behavior.

Extends Uribe's description to non-isodromic Morse polynomial families.

Shows the Principal Poincaré Pontryagin Function can be an iterated integral of length at most 2.

Abstract

It is known that the Principal Poincar\'e Pontryagin Function is generically an Abelian integral. We give a sufficient condition on monodromy to ensure that it is an Abelian integral also in non generic cases. In non generic cases it is an iterated integral. Uribe [17, 18] gives in a special case a precise description of the Principal Poincar\'e Pontryagin Function, an iterated integral of length at most 2, involving logarithmic functions with only one ramification at a point at infinity. We extend this result to some non isodromic families of real Morse polynomials.