Hyperelliptic addition law

TL;DR

This paper derives an explicit addition law for hyperelliptic Abelian functions, providing a fundamental tool for computations in hyperelliptic function theory and algebraic geometry.

Contribution

It presents a new explicit formula for the addition law of hyperelliptic Abelian functions, expanding the computational framework for these functions.

Findings

Explicit addition law for hyperelliptic Abelian functions derived

Basis of the field expressed in terms of $ ext{wp}$ and $ ext{wp'}$ functions

Facilitates algebraic and geometric computations involving hyperelliptic functions

Abstract

We construct an explicit form of the addition law for hyperelliptic Abelian vector functions $\wp$ and $\wp'$. The functions $\wp$ and $\wp'$ form a basis in the field of hyperelliptic Abelian functions, i.e., any function from the field can be expressed as a rational function of $\wp$ and $\wp'$.