Polynomial vector fields in $\mathbb{C}^\infty$ determining differentiation of hyperelliptic functions of any genus

TL;DR

This paper provides explicit proofs for polynomial vector fields in infinite-dimensional complex space, showing their commutativity and their role in differentiating hyperelliptic functions of any genus.

Contribution

It introduces direct proofs for properties of polynomial vector fields related to hyperelliptic functions, expanding understanding of their structure and differentiation.

Findings

Operators commute, ensuring integrability.

Vector fields annul specific polynomials in generating functions.

Results apply to hyperelliptic functions of any genus.

Abstract

In this work we give direct proofs of two theorems concerning explicitly defined polynomial vector fields connected to differentiation of hyperelliptic functions of any genus. We prove that the operators determining the fields commute, and we show that each of them annul polynomials defined in terms of generating functions in $\mathbb{C}^\infty$.