G-adic expansion
Definition
Let \(F, G \in \mathbb{K}((x))[y]\), with \(G \ne 0\). Let \(n = deg_y G\). Since in \(\mathbb{K}((x))[y]\) we have division algorithm, we can write \(F = QG + R\) where \(deg_y R < n\). If \(deg_y Q \ge n\), then write \(Q = Q^1 G + R^1\) with \(deg_y R^1 < n\). With this we can write \(F\) as \(F = Q^1 G^2 + R^1 G + R\). Then we restart with \(Q^1\). This process will stop giving the following expression in terms of the powers of \(G\):
\[ F = \sum_{k = 0}^l \alpha_k(x,y) G^k, \]
where \(deg_y \alpha(x. y) < n\) for all \(k \in \{0, 1, \ldots, l\}\). It is defined the \(G-\)adic expansion of \(F\) as the previous expression of \(F\) in terms of \(G^k\).
Examples
\(\circ\) Let \(F = 3xy^4 + (6x + 1)y^3 + (6x^2 + 4x + 2)y^2 + (6x^2 + 2x + 3)y + 3x^3 + x^2 + x\) and \(G = y^2 + y + 1\). If we compute the first division,
\[ F(x,y) = [3xy^2 + (3x+1)y + (3x^2 + x + 1)]G + 2y = QG + R, \]
and \(deg_y Q = 2 = deg_y G\). Repeating the process,
\[ Q = 3xG + (x + y + 1) = Q^1G + R^1, \]
and \(deg_y Q^1 = 1 < deg_y G\). Therefore, the \(G-\)adic expansion of \(G\) is
\[ F = QG + R = (Q^1G + R^1)G + R = Q^1G^2 + R^1G + R \]
\[ = 3xG^2 + (x + y + 1)G + 2y = \alpha_2(x,y)G^2 + \alpha_1(x,y)G + \alpha_0(x,y). \]
\(\circ\) Let \(F = \sum_{k = 0}^l \alpha_k(x,y) G^k\). It can be proven by induction on \(l \in \mathbb{N}\) that the \(G-\)adic expression is unique.