Computing the fixing group of a rational function

In this paper we develop an algorithm for computing the automorphism group of an intermediate field in the extension K ⊆ K(x). By the classical Lüroth’s theorem any intermediate field F between K and K(x) is of the form F = K(f ) for some rational function f ∈ K(x), see [3,5] and for a constructive proof [2]. Thus, this computational problem is equivalent to determine the fixing group G f of a univariate rational function f . We also present an algorithm for computing F ix(H), the fixed field by a subgroup H ⊆ Aut K K(x). Again, this computational problem is equivalent to finding a Lüroth’s generator of the field fixed by the given subgroup H. Both algorithms are in polynomial time if the field K has a polynomial time algorithm for computing the set of the roots of a univariate polynomial.

The algorithm for computing the fixing group of a rational function uses several techniques related to the rational function decomposition problem. This problem can be stated as follows: given f ∈ K(x), determine whether there exists a decomposition (g, h) of f , f = g(h), with g and h of degree greater than one, and in the affirmative case, compute one. When such a decomposition exists some problems become simpler: for instance, the evaluation of a rational function f can be done with fewer arithmetic operations, the equation f (x) = 0 can be solved more efficiently, improperly parametrized algebraic curves can be reparametrized properly, etc. see [8], [1] and [6]. In fact, a motivation of this paper is to obtain results on rational functional decomposition. As a consequence of our study of G f we provide new and interesting conditions of decomposability of rational functions. Another application of this paper is to study the number m of indecomposable components of a rational function

which is strongly related to the group G f , see [7].

The other algorithm presented for computing the field F ix(H) is based on Galois theory results and the constructive proof of Lüroth’s theorem.

The paper is divided in four sections. In Section 2 we introduce our notations and the background of rational function decomposition. Section 3 studies the Galois group of K(x) over K, the fixing group G f and the field F ix(H), including general theoretical results, and its relation with the functional decomposition problem. Section 4 presents algorithms for computing the fixing group and fixed field. We also give in this section examples illustrating our algorithms.

The set of all non-constant rational functions is a semigroup with identity x, under the element-wise composition of rational functions (symbol • for composition): i.e., given non-constant rational functions g, h ∈ K(x), g • h = g(h).

The units of this semigroup are of the form ax+b cx+d . We will identify these units with the elements of the Galois group of K(x) over K. We will denote this group by Γ(K) = Aut K K(x).

Given f ∈ K(x), we will denote as f N , f D the numerator and denominator of f respectively, assuming that f N and f D are relatively prime. We define the degree of f as deg

If g, h ∈ K(x) are rational functions of degree greater than one, f = g • h = g(h) is their (functional) composition, (g, h) is a (functional) decomposition of f , and f is a decomposable rational function.

The following lemma describes some basic properties of rational function decomposition, see [1] for a proof. Theorem 1. With the above notations and definitions, we have the following:

• The units with respect to composition are precisely the rational functions u with deg u = 1.

Furthermore, it can be computed from f and h by solving a linear system of equations.

for some g ∈ K(x). From this fact the following natural concept arises:

The next result is an immediate consequence of Lüroth’s theorem.

Corollary 1. Let f ∈ K(x) be a non-constant rational function. Then the equivalence classes of the decompositions of f correspond bijectively to intermediate fields F, K(f ) ⊆ F ⊂ K(x).

3 The Galois Correspondences in the Extension K ⊆ K(x).

We start defining our main notions and tools.

Definition 2. Let K be any field.

• Let H be a subgroup of Γ(K). The fixed field by H is Fix(H),

Before we discuss the computational aspects of these concepts, we will need some properties based on general facts from Galois theory. Theorem 2.

• Given a finite subgroup H of Γ, there is a bijection between the subgroups of H and intermediate fields in

there is a bijection between the right components of f (up to equivalence by units) and the subgroups of H.

• Let f ∈ K(x) and u, v be two units. Let H < Γ(K)

Unfortunately, it is not true that [K(x) : K(f )] = |G f |. However, some interesting results about decomposability can be given. Theorem 3. Let f be indecomposable.

According of above theorem if H is infinite then F ix(H) is trivial. Some t

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