Algebraic independence of multipliers of periodic orbits in the space of polynomial maps of one variable (Q2813944)

The author studies the space of complex polynomials of degree \(n\geq 3\) in the moduli space with \(n -1\) distinct marked periodic orbits of given periods. ``The question about algebraic independence of multipliers of periodic orbits in the polynomial case was asked by Ilyashenko in the context of studying the Kupka-Smale property for volume preserving polynomial automorphisms of \(\mathbb{C}^{2}\)'' [\textit{I. Gorbovicks}, Some problems from complex dynamical systems and combinatorial geometry. Ithaca, NY: Cornell University (PhD Thesis) (2012)]. The author's main result answers to this question when the number of multipliers is less than \(n\). The author proves that this space is an irreducible algebraic set and that the multipliers of the marked periodic orbits are algebraically independent over \(\mathbb{C}\). The author also gives a similar result about independence of multipliers for a certain class of affine subspaces of the space of complex polynomials of degree \(n\).