Approximate root
Definition
Let \(f(x,y) = y^n + a_1(x)y^{n-1} + \cdots + a_n(x) \in \mathbb{K}((x))[y]\) be an irreducible polynomial. Let us consider \(d\) a divisor of \(n\) and let \(G \in \mathbb{K}((x))[y]\) be a monic polynomial of degree \(\frac{n}{d}\) in \(y\). Let
\[ f = G^d + \alpha_1 G^{d-1} + \cdots + \alpha_d, \]
be the G-adic expansion of \(f\). It is said that \(G\) is a \(d^{th}\) approximate root of \(f\) if \(\alpha_1 = 0\).
It can be proven that for any \(d\) divisor of \(n\), the \(d^{th}\) approximate root of \(f\) exists and it is unique. It is denoted by \(Ap(f;d)\).
Examples
\(\circ\) Let \(f = y^4 + 13x^2y^2 + 42x^2\), and \(d = 2\), let us calculate \(Ap(f; 2)\) applying the following algorithm. First, let us consider \(G = y^{\frac{n}{d}} = y^2\) and compute the \(G-\)adic expansion of \(f\),
\[ f = y^4 + 13x^2y^2 + 42x^2 = G^2 + 13x^2G + 42x^2 \]
We have \(\alpha_1(x,y) = 13x^2 \ne 0\), then, we consider now \(G_1 = T(G) = G + \frac{\alpha_1(x,y)}{d} = y^2 + \frac{13}{2}x^2\), where \(T(G)\) is the Tschirnhausen transform. If we compute the \(G_1-\)adic expansion of \(f\),
\[ f = G^2 + 13x^2G + 42x^2 = (G_1 - \frac{13}{2}x^2)^2 + 13x^2(G_1 - \frac{13}{2}x^2) + 42x^2 \]
\[ = G_1^2 - 13x^2G_1 + \frac{169}{4}x^4 + 13x^2G_1 - \frac{169}{2}x^4 + 42x^2 = G_1^2 - \frac{169}{4}x^4 + 42x^2. \] and \(\alpha_1(x,y) = 0\).
To sum up, \(Ap(f;2) = y^2 + \frac{13}{2}x^2\).