Differentiation of Genus 3 Hyperelliptic Functions

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 $p:{\mathbb{C}}^{3g}\to {\mathbb{C}}^{2g}$. 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.

15 pages

References in corpus (2)

• Multi-Dimensional Sigma-Functions
• Polynomial dynamical systems and Korteweg--de Vries equation

Cited by in corpus (5)

• Parametric Korteweg--de Vries hierarchy and hyperelliptic sigma functions
• Differentiation of Genus $4$ Hyperelliptic Functions
• Sigma functions and Lie algebras of Schrödinger operators
• The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus 3 and applications
• Hyperelliptic sigma functions and Adler-Moser polynomials