Computing Borcherds Products

TL;DR

This paper introduces an efficient polynomial-time algorithm for computing Borcherds products, overcoming exponential runtime issues of naive methods by effectively managing Fourier expansion bounds, with practical implementation demonstrating significant speed improvements.

Contribution

The paper presents a novel polynomial-time algorithm for computing Borcherds products, improving efficiency over previous exponential-time approaches.

Findings

Algorithm has polynomial runtime

Implementation shows practical speedup

Handles Fourier bounds efficiently

Abstract

We present an algorithm for computing Borcherds products, which has polynomial runtime. It deals efficiently with the bounds on Fourier expansion indices originating in Weyl chambers. Naive multiplication has exponential runtime due to inefficient handling of these bounds. An implementation of the new algorithm shows that it is also much faster in practice.