An algorithm to compute the differential equations for the logarithm of a polynomial

TL;DR

This paper introduces an algorithm to compute the differential equations governing the logarithm of a polynomial, facilitating the analysis of integrals involving logarithmic functions within the framework of D-module theory.

Contribution

The paper presents a novel algorithm for computing the annihilator of the logarithm of a polynomial, extending existing methods to include logarithmic factors and their holonomic systems.

Findings

Algorithm effectively computes differential equations for log-polynomial functions.

Enables analysis of integrals involving logarithmic polynomials using D-module techniques.

Extends the class of functions with computable holonomic systems.

Abstract

We present an algorithm to compute the annihilator of (i.e., the linear differential equations for) the logarithm of a polynomial in the ring of differential operators with polynomial coefficients. The algorithm consists of differentiation with respect to the parameter s of the annihilator of f^s for a polynomial f and quotient computation. More generally, the annihilator of f^s(log f)^m for a complex number s and a positive integer m can be computed, which constitutes what is called a holonomic system in D-module theory. This enables us to compute a holonomic system for the integral of a function involving the logarithm of a polynomial by using integration algorithm for D-modules.