Tschirnhausen transform
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 \(y\). Then, the G-adic expansion of \(f\) has the form
\[ f = G^d + \alpha_1 G^{d-1} + \cdots + \alpha_d, \]
with \(\alpha_i \in \mathbb{K}((x))[y]\) and \(deg_y \alpha_i(x,y) < \frac{n}{d}\) for all \(i \in \{1, 2, \ldots, d\}\). It is defined the Tschirnhausen transform of \(G\) as \(T(G) = G + \frac{\alpha_1}{d}\). This definition is related with the concept of approximate root of an irreducible polynomial.
Examples
\(\circ\) Let
\[ f = y^6 + (12x^2 + 3)y^5 + (36x^4 + 18x^2 + 2)y^4 + 13xy^3 + (78x^3 + 2x^2 + 20x)y^2 + 42x^2, \]
and \(g = y^3 + (6x^2 + 1)y^2 + 7x\). We have \(n = 6\) and \(d = 2\). Dividing,
\[ f = (y^3 + (6x^2 + 2)y^2 + 6x)G + 2x^2y^2 = QG + R, \]
and \(deg_y Q = 3 = deg_y G\). Dividing \(Q\) by \(G\),
\[ Q = G + y^2 - x = Q^1G + R^1. \]
Then, the \(G-\)adic expansion of \(f\) is
\[ f = QG + R = (Q^1G + R^1)G + R = G^2 + (y^2 - x)G + 2x^2y^2, \]
with \(\alpha_1(x,y) = y^2 - x\). Finally, the Tschirnhausen transform of \(G\) is
\[ T(G) = G + \frac{\alpha_1(x,y)}{d} = y^3 + (6x^2 + 1)y^2 + 7x + \frac{y^2-x}{2}. \]