On the functional decomposition of multivariate laurent polynomials (abstract only)

Abstract

Determining whether a univariate polynomial may be written as the functional composition of two others of lower degree is a question that has been studied since at least the work of Ritt [1]. Algorithms by Barton and Zippel [2] and then by Kozen and Landau [3] have been incorporated in many computer algebra systems. Generalizations have been studied for functional decomposition of rational functions [4], algebraic functions [5], multivariate polynomials [6] and univariate Laurent polynomials [7]. We explore the functional decomposition problem for multivariate Laurent polynomials, considering the case f = g o h where g is univariate and h may be multivariate. We present an algorithm to find such a decomposition if it exists. The algorithm proceeds as follows: First, a variable weighting is chosen to make the weighted degree zero term in f constant. The positive degree and negative degree parts of h are then reconstructed separately, in a manner similar to that of Kozen and Landau, but by treating the homogeneous collections of terms by grade rather than individual monomials. Then terms of the univariate polynomial g are reconstructed degree by degree using a generic univariate projection of h.