Towards computing vector partition functions by iterated partial fraction decomposition

TL;DR

This paper explores a method using iterated partial fraction decomposition to compute vector partition functions, providing an algorithm for formal numerator computation and analyzing the types of rational functions involved.

Contribution

It introduces an algorithm for formal numerator computation in partial fraction decompositions of rational functions related to vector partition functions.

Findings

Algorithm successfully computes numerators in partial fraction decompositions.

Analysis of generalized rational functions encountered during the decomposition process.

Provides a framework for handling complex rational functions in the context of vector partition functions.

Abstract

We investigate the possibilities to calculate vector partition functions by means of iterated partial fraction decomposition, as suggested by Beck (2004). Particularly, for an important type of families of rational functions, we describe an algorithm to compute the numerators in their partial fraction decomposition "formally," i.e., as a formal expression in the parameter. We also analyze the type of generalized rational functions that appear during the execution of an algorithm based on iterated partial fraction decomposition and explain how to handle these objects.