Communications in Number Theory and Physics

Volume 13 (2019)

Number 2

Rationalizing roots: an algorithmic approach

Pages: 253 – 297

DOI: https://dx.doi.org/10.4310/CNTP.2019.v13.n2.a1

Authors

Marco Besier (Institut für Mathematik, Johannes Gutenberg-Universität, Mainz, Germany; and PRISMA Cluster of Excellence, Institut für Physik, Johannes Gutenberg-Universität, Mainz, Germany)

Duco van Straten (Institut für Mathematik, Johannes Gutenberg-Universität, Mainz, Germany)

Stefan Weinzierl (PRISMA Cluster of Excellence, Institut für Physik, Johannes Gutenberg-Universität, Mainz, Germany)

Abstract

In the computation of Feynman integrals which evaluate to multiple polylogarithms one encounters quite often square roots. To express the Feynman integral in terms of multiple polylogarithms, one seeks a transformation of variables, which rationalizes the square roots. In this paper, we give an algorithm for rationalizing roots. The algorithm is applicable whenever the algebraic hypersurface associated with the root has a point of multiplicity $(d-1)$, where $d$ is the degree of the algebraic hypersurface. We show that one can use the algorithm iteratively to rationalize multiple roots simultaneously. Several examples from high-energy physics are discussed.

Received 1 October 2018

Accepted 5 December 2018

Published 26 April 2019