Skew-invariant curves and the algebraic independence of Mahler functions

Abstract

For $p\in {\mathbb{Q}}_{+}\setminus \{1\}$ 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 $g_1,\dots ,g_n$ each satisfy some difference equation with respect to the substitution $t\mapsto t^q$, then $f$ is algebraically independent from $g_1,\dots ,g_n$. These theorems are themselves consequences of a refined classification of skew-invariant curves for split polynomial dynamical systems on ${\mathbb{A}}^2$.