  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 $\mathbb{A}^2$.
