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 $\mathbb{A}^2$.
The range of multiplicative functions on C[x], R[x] and Z[x]
Mahler's measure is generalized to create the class of { \it multiplicative distance functions}. These functions measure the complexity of polynomials based on the location of their zeros in the complex plane . Following work of S.-J. Chern and J. Vaaler in \cite{chern-vaaler} , we associate to each multiplicative distance function two families of analytic functions which encode information about its range on \C [x] and \R [x]. These { \it moment functions} are Mellin transforms of distribution functions associated to the multiplicative distance function and demonstrate a great deal of arithme…
