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Moscow Mathematical Journal

Volume 13, Issue 4, October–December 2013  pp. 555–583.

Inductive Solution of the Tangential Center Problem on Zero-Cycles

Authors A. Álvarez (1), J.L. Bravo (1), and P. Mardešić (2)
Author institution: (1) Departamento de Matemáticas, Universidad de Extremadura, Avenida de Elvas s/n, 06006 BADAJOZ Spain
(2) Université de Bourgogne, Institut de Mathématiques de Bourgogne, UMR 5584 du CNRS, UFR Sciences et Techniques, 9, av. A. Savary, BP 47870, 21078 DIJON CEDEX, France


Abstract. Given a polynomial f ∈ ℂ[z] of degree m, let z1(t), …, zm(t) denote all algebraic functions defined by f(zk(t)) = t. Given integers n1, … , nm such that n1+⋯+nm = 0, the tangential center problem on zero-cycles asks to find all polynomials g ∈ ℂ[z] such that n1g(z1(t)) + ⋯ + nm g(zm(t)) ≡ 0. The classical center-focus problem, or rather its tangential version in important non-trivial planar systems lead to the above problem.

The tangential center problem on zero-cycles was recently solved in a preprint by Gavrilov and Pakovich.

Here we give an alternative solution based on induction on the number of composition factors of f under a generic hypothesis on f. First we show the uniqueness of decompositions f = f1◦⋯◦fd such that every fk is 2-transitive, monomial or a Chebyshev polynomial under the assumption that in the above composition there is no merging of critical values.

Under this assumption, we give a complete (inductive) solution of the tangential center problem on zero-cycles. The inductive solution is obtained through three mechanisms: composition, primality and vanishing of the Newton–Girard component on projected cycles.

2010 Mathematics Subject Classification. 34C07, 34C08, 34M35, 14K20

Keywords:  Abelian integrals, tangential center problem, center-focus problem, moment problem

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