25 - An application of Gröbner Bases to the Decomposition of Rational Mappings

Summary

Abstract

Let k(X) ≔ Quot(k[x₁,…, x_n]/I(X)) be the field of rational functions of an irreducible affine variety X. The decomposition f = g o h of a rational mapping f ∈ k(X)^m is defined in terms of subfields of k(X). Taking a decomposition g o h of f for a strategy to solve equations of the form f(x) = α in several steps the definition is extended to the “inverse” mappings f^-1 : α → f^-1(α).

To compute the “inner part” h of a decomposition g o h intermediate fields of k(f) and k(X) have to be found; e.g., for k(X)/k(f) separable algebraic the coefficients of suitable Gröbner bases can be used for performing this task. To compute the “outer part” g a solution to the field membership problem for subfields of k(X) by means of Gröbner basis techniques is suggested which does not make use of tag variables. Finally “monomial decompositions” are considered; in this case the required Gröbner basis computations can be performed efficiently.

Introduction

The problem of decomposing a function f into functions g and h such that f = g o h is the functional composition hereof has been studied by various authors, as Dickerson (1989) for the case when f is a multivariate polynomial over a field, Zippel (1991) for f a univariate rational function or Kozen et al. (1996) for f a univariate algebraic function.