  Let $M=(m_{i})_{i=0}^{\infty}$ be a sequence of integers such that $m_{0}=1$
and $m_{i}\geq 2$ for $i\geq 1$. In this paper we study $M$-ary partition
polynomials $(p_{M}(n,t))_{n=0}^{\infty}$ defined as the coefficient in the
following power series expansion: \begin{align*}
\prod_{i=0}^{\infty}\frac{1}{1-tq^{M_{i}}} = \sum_{n=0}^{\infty}
p_{M}(n,t)q^{n}, \end{align*} where $M_{i}=\prod_{j=0}^{i}m_{j}$. In
particular, we provide a detailed description of their rational roots and show,
that all their complex roots have absolute values not greater than $2$. We also
study arithmetic properties of $M$-ary partition polynomials. One of our main
results says that if $n=a_{0}+a_{1}M_{1}+\cdots +a_{k}M_{k}$ is a (unique)
representation such that $a_{j}\in\{0,\ldots ,m_{j+1}-1\}$ for every $j$, then
\begin{align*} p_{M}(n,t)\equiv t^{a_{0}}\prod t^{a_{j}}f(a_{j}+1,t^{m_{j}-1})
\pmod{g_{k}(t)}, \end{align*} where $f(a,t):=\frac{t^{a}-1}{t-1}$ and
$g_{k}(t):=\gcd \big(t^{m_{1}+m_{2}-1}f(m_{2},t^{m_{1}-1}),\ldots
,t^{m_{k}+m_{k+1}-1}f(m_{k+1},t^{m_{k}-1})\big)$. This is a polynomial
generalisation of the well-known characterisation modulo $m$ of the sequence of
$m$-ary partition.
