Differentiation of Genus $4$ Hyperelliptic Functions

TL;DR

This paper provides an explicit method for differentiating hyperelliptic functions specifically in genus 4, extending classical and recent results from elliptic to higher-genus hyperelliptic functions.

Contribution

It offers the first explicit differentiation formula for genus 4 hyperelliptic functions, building on prior solutions for genus 2 and 3 cases.

Findings

Explicit differentiation formula for genus 4 hyperelliptic functions

Extension of classical elliptic function results to genus 4

Provides tools for further research in hyperelliptic function theory

Abstract

In this work we give an explicit solution to the problem of differentiation of hyperelliptic functions in genus $4$ case. It is a genus $4$ analogue of the classical result of F. G. Frobenius and L. Stickelberger [F. G. Frobenius, L. Stickelberger, "Uber die Differentiation der elliptischen Functionen nach den Perioden und Invarianten", J. Reine Angew. Math., 92 (1882), 311-337] in the case of elliptic functions. An explicit solution in the genus $2$ case was given in [V. M. Buchstaber, "Polynomial dynamical systems and Korteweg-de Vries equation", Proc. Steklov Inst. Math., 294 (2016), 176-200]. An explicit solution in the genus $3$ case was given in [E. Yu. Bunkova, "Differentiation of genus 3 hyperelliptic functions", European Journal of Mathematics, 4:1 (2018), 93-112].