Moscow Mathematical Journal

Abstract. Given a polynomial f ∈ ℂ[z] of degree m, let z₁(t), …, z_m(t) denote all algebraic functions defined by f(z_k(t)) = t. Given integers n₁, … , n_m such that n₁+⋯+n_m = 0, the tangential center problem on zero-cycles asks to find all polynomials g ∈ ℂ[z] such that n₁g(z₁(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 = f₁◦⋯◦f_d such that every f_k 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