, Volume 5, Issue 2, pp 285-300

Algebraic closure of a rational function

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Abstract

We give a simple algorithm to decide if a non-constant rational fractionR=P/Q in the field \(\mathbb{K}(x) = \mathbb{K}(x_1 ,...,x_n )\) inn≥2 variables over a fieldK of characteristic 0 can be written as a non-trivial compositionR=U(R 1), whereR 1 is anothern-variable rational fraction whereasU is a one-variable rational fraction which is not a homography.

More precisely, this algorithm produces a generator of thealgebraic closure of a rational fraction in the fieldK x.

Although our algorithm is simple (it uses only elementary linear algebra), its proof relies on a structure theorem: the algebraic closure of a rational fraction is a purely transcendental extension ofK of transcendence degree 1.

Despite this theorem is a generalization of a result of Poincaré about the rational first integrals of polynomial planar vector fields, we found it useful to give a complete proof of it: our proof is as algebraic as possible and thus very different from Poincaré's original work.