Vanishing abelian integrals on zero-dimensional cycles

DOI10.1112/PLMS/PDT012zbMATH Open1279.30045arXiv1101.1777OpenAlexW3099043989MaRDI QIDQ2872028

Amelia Álvarez Sánchez, José Luis Bravo Trinidad, Pavao Mardešić

Publication date: 14 January 2014

Published in: Proceedings of the London Mathematical Society. Third Series (Search for Journal in Brave)

Abstract: In this paper we study conditions for the vanishing of Abelian integrals on families of zero-dimensional cycles. That is, for any rational function

f (z)

, characterize all rational functions

g (z)

and zero-sum integers

n_{i}

such that the function

t m a p s t o s u m n_{i} g (z_{i} (t))

vanishes identically. Here

z_{i} (t)

are continuously depending roots of

f (z) - t

. We introduce a notion of (un)balanced cycles. Our main result is an inductive solution of the problem of vanishing of Abelian integrals when

f, g

are polynomials on a family of zero-dimensional cycles under the assumption that the family of cycles we consider is unbalanced as well as all the cycles encountered in the inductive process. We also solve the problem on some balanced cycles. The main motivation for our study is the problem of vanishing of Abelian integrals on single families of one-dimensional cycles. We show that our problem and our main result are sufficiently rich to include some related problems, as hyper-elliptic integrals on one-cycles, some applications to slow-fast planar systems, and the polynomial (and trigonometric) moment problem for Abel equation. This last problem was recently solved by Pakovich and Muzychuk (cite{PM} and cite{P}). Our approach is largely inspired by their work, thought we provide examples of vanishing Abelian integrals on zero-cycles which are not given as a sum of composition terms contrary to the situation in the solution of the polynomial moment problem.

Full work available at URL: https://arxiv.org/abs/1101.1777

zbMATH Keywords

moment problem abelian integral

Mathematics Subject Classification ID

Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable (30D05) Moment problems and interpolation problems in the complex plane (30E05)