Computing Riemann theta functions in Sage with applications

Abstract

A new implementation for the computation of the Riemann theta function in the open-source mathematical software Sage is discussed. This implementation is used in two applications. The first is the computation of three-phase solutions of the Kadomtsev–Petviashvili equation using an algorithm due to Dubrovin, originally implemented by Dubrovin et al. Our implementation is significantly easier, due to our more straightforward computation of the theta function. The second application is that of the computation of the bitangents of a quartic plane algebraic curve, relevant in convex optimization. Since Sage currently lacks the tools for computing with Riemann surfaces, this second application relies partially on results obtained using Maple's algcurves package. The current manuscript is the first step towards porting the functionality of the algcurves package to Sage as well as other scientific Python distributions.