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

TL;DR

This paper proves that the multipliers of distinguished periodic orbits in polynomial maps of one variable are algebraically independent functions over the complex numbers, revealing a fundamental property of the parameter space.

Contribution

It establishes the algebraic independence of multipliers of periodic orbits in the space of degree n polynomials, a novel result in complex dynamics.

Findings

Multipliers are algebraically independent functions.

The result applies to polynomials of degree n ≥ 3.

Provides new insights into the structure of polynomial parameter spaces.

Abstract

We consider a space of complex polynomials of degree $n\ge 3$ with $n-1$ distinguished periodic orbits. We prove that the multipliers of these periodic orbits considered as algebraic functions on that space, are algebraically independent over the field of complex numbers.