Differentiation of Genus 3 Hyperelliptic Functions

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}$\mathrm{.}For$g = 1,2,3$wedescribe$3g$polynomialvectorfieldsin$$\mathbb{C}$^{3g}$projectablefor$p$ and their polynomial Lie algebras. We obtain the corresponding derivations of the field of hyperelliptic functions.