Vanishing Abelian integrals on zero‐dimensional cycles

• Published in: Proceedings of the London Mathematical Society 2013-12, Vol.107 (6), p.1302-1330
• Main Authors: Álvarez, A., Bravo, J. L., Mardešić, P.
• Format: Article
• Language: English
• Subjects: Balancing; Functions (mathematics); Integers; Integrals; Mathematical analysis; Polynomials; Rational functions; Roots
• ISSN: 0024-6115; 1460-244X
• DOI: 10.1112/plms/pdt012

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 {ni} such that the function t↦∑ nig(zi(t)) vanishes identically. Here zi(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 as well as all the cycles encountered in the inductive process is unbalanced. We also solve the problem on some balanced cycles. The main motivation for our study is the problem of the 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, such as hyper‐elliptic integrals on one‐cycles, some applications to slow‐fast planar systems and the polynomial (and trigonometric) moment problem for the Abel equation. This last problem was recently solved by Pakovich [Bull. Sci. Math. 133 (2009) 693–732] and Pakovich and Muzychuk [Proc. London Math. Soc. 99 (2009) 633–657]. Our approach is largely inspired by their work, thought we provide examples of vanishing Abelian integrals on zero‐cycles that are not given as a sum of composition terms in contrast to the situation in the solution of the polynomial moment problem.