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.
Marc Moreno Maza
2008-03-18