Algebraic solution of the Jacobi inverse problem and explicit addition laws

TL;DR

This paper presents an algebraic solution to the inverse Jacobi problem and derives explicit addition laws on algebraic curves, extending previous work and providing a coordinate-free approach.

Contribution

It introduces a new algebraic method for solving the inverse Jacobi problem and generalizes addition laws on algebraic curves in a coordinate-free manner.

Findings

Explicit addition laws on algebraic curves derived

Solution generalizes previous laws by Leykin

Addresses a question by T. Shaska about coordinate-free laws

Abstract

We formulate a solution to the Algebraic version of the Inverse Jacobi problem. Using this solution we produce explicit addition laws on any algebraic curve generalizing the law suggested by Leykin [2] in the case of (n, s) curves. This gives a positive answer to a question asked by T. Shaska whether addition laws appearing in [2] can be produced in a coordinate free manner.