# 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

**Summary:**

Abstract. Given a polynomial *f* ∈ ℂ[*z*] of degree *m*, let *z*_{1}(*t*), …, *z _{m}*(

*t*) denote all algebraic functions defined by

*f*(

*z*(

_{k}*t*)) =

*t*. Given integers

*n*

_{1}, … ,

*n*such that

_{m}*n*

_{1}+⋯+

*n*= 0, the tangential center problem on zero-cycles asks to find all polynomials

_{m}*g*∈ ℂ[

*z*] such that

*n*

_{1}

*g*(

*z*

_{1}(

*t*)) + ⋯ +

*n*(

_{m}g*z*(

_{m}*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* = *f*_{1}◦⋯◦*f _{d}* such that
every

*f*is 2-transitive, monomial or a Chebyshev polynomial under the assumption that in the above composition there is no merging of critical values.

_{k}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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