Differentiation of genus 3 hyperelliptic functions

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 Frobenius and Stickelberger (J Reine Angew Math 92:311-337, 1882). Our method is based on the series of works by Buchstaber, Enolskii and Leikin. First we introduce a polynomial map p: C-3g -> C-2g. Next for g = 1, 2, 3 we provide 3g polynomial vector fields in C-3g projectable for p and describe their polynomial Lie algebras. Finally we obtain the corresponding derivations of the field of hyperelliptic functions.