On Polynomial Decompositions

The purpose of this paper is to introduce the norm decomposition which enables us to compute the roots of a monic irreducible imprimitive polynomial f ∈ Q[t] by solving polynomial equations of lower degree. We call an irreducible polynomial f imprimitive if the number field generated by a root of f contains non-trivial subfields. We will see that for each subfield there exists a norm decomposition. The norm decomposition generalizes the functional (Kozen and Landau, 1989) and homogeneous bivariate decompositions (von zur Gathen and Weiss, 1995). There exist imprimitive polynomials having neither a functional nor a homogeneous bivariate decomposition, but which do have a norm decomposition. Furthermore, the computing times by our algorithm are much shorter than the ones for a homogeneous bivariate decomposition. If a functional decomposition f = g(h) with g, h ∈ Q[t] exists we can calculate the roots β1, . . . , βm of g, and then the roots of h−βi (1 ≤ i ≤ m) in order to obtain the roots of f . Note that there are very efficient algorithms to compute functional decompositions. In the homogeneous bivariate decomposition, the polynomial f is written in the form f = ĝ(h1, h2) where ĝ ∈ Q[t, u] is homogeneous and h1, h2 ∈ Q[t]. A drawback of the known algorithms for computing a homogeneous bivariate decomposition is that they require an expensive factorization of the polynomial f in K[t], where K is the number field generated by a root of f . If f has a homogeneous bivariate decomposition then f = h2 g( h1 h2 ), where g(t) = ĝ(t, 1) and m = deg(ĝ). As f is irreducible we obtain the roots of f by first computing the roots β1, . . . , βm of g and then the roots of the polynomials h1 − βih2 (1 ≤ i ≤ m). It is well known that the existence of subfields Q(β) ⊆ Q(α) (Dixon, 1990; Lazard and Valibouze, 1993; Hulpke, 1995; Casperson et al., 1996; Klüners and Pohst, 1997) is equivalent to f | g(h) where f, g, h ∈ Q[t] and f , g are the minimal polynomials of α and β, respectively. This is a generalization of the functional decomposition. Lazard and Valibouze (1993) illustrate by an example how to represent the roots of f by a “nested” system of equations which can be obtained via computing subfields.