On Multivariate Rational Function Decomposition

If K is a field, and g, h ∈ K(x) are rational functions of degree greater than one, then 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 univariate rational functional decomposition problem can be stated as follows: given f ∈ K(x), determine whether there exists a decomposition (g, h) of f 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 more efficiently solved, improperly parametrized algebraic curves can be reparametrized properly, etc. Zippel (1991) presented a polynomial time algorithm to decompose a univariate rational function over any field with efficient polynomial factorization. Alonso et al. (1995) presented two exponential-time algorithms to decompose univariate rational functions, which are quite efficient in practice. Klüners (2000) presented an exponential-time algorithm to decompose univariate rational functions over Q. If f, h ∈ K(x) are such that K(f) ⊂ K(h) ⊂ K(x), then f = g(h) for some g ∈ K(x). By the classical Lüroth’s theorem (see Lüroth, 1876) this problem can be translated into field theory: given f ∈ K(x) compute, if it exists, a proper intermediate field F such that K(f) ⊂ F ⊂ K(x). The following extended version of Lüroth’s theorem is a central result, as it allows to generalize this problem to multivariate rational functions.