On the computation of fundamental functions and Abelian differentials of the third kind
Yu Ying, E. A. Ayryan, M. D. Malykh, L. A. Sevastianov
TL;DR
The paper addresses computing the fundamental function and Abelian differentials of the third kind on smooth plane curves over $\mathbb{C}$, emphasizing the computational challenges posed by algebraic number fields. It describes implementing Weierstrass's algorithm in Sage and introduces a symmetrization of the third-kind differential construction to avoid costly direct linear systems over $\overline{\mathbb{Q}}$. A key result is that the symmetrized linear systems reduce to rational-coefficient subsystems, enabling practical computation in examples such as elliptic curves, and it discusses the duality between the fundamental function and third-kind differentials. The work lays the groundwork for computing the fundamental function via this duality and outlines future integration with Sage's Symmetric Functions to further simplify expressions, enhancing the feasibility of these classical constructions in computer algebra systems.
Abstract
We consider the construction of the fundamental function and Abelian differentials of the third kind on a plane algebraic curve over the field of complex numbers that has no singular points. The algorithm for constructing differentials of the third kind is described in Weierstrass's Lectures. The article discusses its implementation in the Sage computer algebra system. The specificity of this algorithm, as well as the very concept of the differential of the third kind, implies the use of not only rational numbers, but also algebraic ones, even when the equation of the curve has integer coefficients. Sage has a built-in algebraic number field tool that allows implementing Weierstrass's algorithm almost verbatim. The simplest example of an elliptic curve shows that it requires too many resources, going far beyond the capabilities of an office computer. Then the symmetrization of the method is proposed and implemented, which solves the problem and allows significant economy of resources. The algorithm for constructing a differential of the third kind is used to find the value of the fundamental function according to the duality principle. Examples explored in the Sage system are provided.