Naslov
Inductive solution of the tangential center problem on zero-cycles.

Autori
Álvarez, Amelia. ; Bravo, Jose. Luis. ; Mardešić, Pavao

Izvornik
Moscow Mathematical Journal (1609-3321) 13 (2013), 4; 555-583

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
center problem; tangential center; cycle

Sažetak
Given a polynomial f from C[z] of degree m, let z1(t), ..., zm(t) denote all algebraic functions de_ned by f(zk(t)) = t. Given integers n1, ..., nm such that n1 +...+nm = 0, the tangential center problem on zero-cycles asks to _nd all polynomials g 2 C[z] such that n1g(z1(t)) + : : : + nmg(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 [14]. 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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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