Skew-invariant curves and the algebraic independence of Mahler functions

Abstract

For $p \in \mathbb{Q}_+ \smallsetminus \{ 1 \}$ a positive rational number different from one, we say that the Puisseux series $f \in \mathbb{C}((t))^\text{alg}$ is $p$-Mahler of non-exceptional polynomial type if there is a polynomial $P \in \mathbb{C}(t)^\text{alg}[X]$ of degree at least two which is not conjugate to either a monomial or to plus or minus a Chebyshev polynomial for which the equation $f(t^p) = P(f(t))$ holds. We show that if $p$ and $q$ are multiplicatively independent and $f$ and $g$ are $p$-Mahler and $q$-Mahler, respectively, of non-exceptional polynomial type, then $f$ and $g$ are algebraically independent over $\mathbb{C}(t)$. This theorem is proven as a consequence of a more general theorem that if $f$ is $p$-Mahler of non-exceptional polynomial type, and $g_1, \ldots, g_n$ each satisfy some difference equation with respect to the substitution $t \mapsto t^q$, then $f$ is algebraically independent from $g_1, \ldots, g_n$. These theorems are themselves consequences of a refined classification of skew-invariant curves for split polynomial dynamical systems on $\mathbb{A}^2$.