# Differentiation of Genus 3 Hyperelliptic Functions

## TL;DR

This paper provides an explicit method for differentiating genus 3 hyperelliptic functions, extending classical results and utilizing polynomial maps and vector fields to derive derivations of hyperelliptic function fields.

## Contribution

It introduces a new explicit differentiation framework for genus 3 hyperelliptic functions, generalizing previous genus 1 and 2 results and employing polynomial vector fields and Lie algebras.

## Key findings

- Explicit polynomial map for genus 3 hyperelliptic functions
- Polynomial vector fields projectable under the map
- Derivations of hyperelliptic function fields obtained

## Abstract

In this work we give an explicit solution to the problem of differentiation of hyperelliptic functions in genus $3$ case. It is a genus $3$ analogue of the result of F. G. Frobenius and L. Stickelberger. Our method is based on the series of works by V. M. Buchstaber, D. V. Leikin and V. Z. Enolskii. We describe a polynomial map $p\colon \mathbb{C}^{3g} \to \mathbb{C}^{2g}$. For $g = 1,2,3$ we describe $3g$ polynomial vector fields in $\mathbb{C}^{3g}$ projectable for $p$ and their polynomial Lie algebras. We obtain the corresponding derivations of the field of hyperelliptic functions.