Algorithms for the Functional Decomposition of Laurent Polynomials

Recent work has detailed the conditions under which univariate Laurent polynomials have functional decompositions. This paper presents algorithms to compute such univariate Laurent polynomial decompositions efficiently and gives their multivariate generalization. One application of functional decomposition of Laurent polynomials is the functional decomposition of so-called “symbolic polynomials.” These are polynomial-like objects whose exponents are themselves integer-valued polynomials rather than integers. The algebraic independence of X, X, X 2/2, etc, and some elementary results on integer-valued polynomials allow problems with symbolic polynomials to be reduced to problems with multivariate Laurent polynomials. Hence we are interested in the functional decomposition of these objects.