Statement

Let $a$ and $b$ be two univariate polynomials in ${\mbox{${\mathbb{Q}}$}}[x]$ , with respective degrees $m$ and $n > 0$ . We call $b$ -adic expansion of $a$ , a sequence of polynomials $c_{k}, \ldots, c_1, c_0$ in ${\mbox{${\mathbb{Q}}$}}[x]$ such that

1. $a = c_k b^k + \cdots + c_1 b + c_0$ ,
2. the degree of each $c_{k}, \ldots, c_1, c_0$ is strictly less than that of $b$ ,
3. $c_k = 0$ holds if and only $a$ is null.

This is an adaptation to polynomials of a well-known notion for integer numbers.