Pseudo-approximate root of an irreducible polynomial
Definition
Let \(f(x,y) \in \mathbb{K}((x))[y]\) irreducible and \(y(t) = \sum_{p \in \mathbb{N}} c_p t^p \in \mathbb{K}((t))\) a root of \(f(t^n, y(t)) = 0\). Let \(\underline{m} = (m_0, m_1, \ldots, m_h)\), \(\underline{d} = (d_1, d_2, \ldots, d_{h+1})\) and \(\underline{r} = (r_0, r_1, \ldots, r_h)\) the characteristic sequences of \(f\). For all \(k \in \{1, 2, \ldots, h\}\), let \(\overline{y}(t) = \sum_{p < m_k} c_p t^p\) and \(Y(t) = \overline{y}(t^{1/d_k}) \in \mathbb{K}((t))\). It is defined the \(d_k^{th}-\)pseudo-approximate root of \(f\), denoted by \(G_k(x,y)\), as the minimal polynomial of \(Y(t)\) over \(\mathbb{K}((t^{n/d_k}))\).
It can be proven that the \(d_k^{th}-\)pseudo-approximate root of \(f\) is of the form
\[ G_k(t^{\frac{n}{d_k}}, y) = \prod_{v, v^{n/d_k} = 1} (y - Y(vt)). \]