Polynomial vector fields in $\mathbb{C}^\infty$ determining differentiation of hyperelliptic functions of any genus
E. Yu. Bunkova
TL;DR
The paper addresses constructing polynomial vector fields in $\mathbb{C}^{\infty}$ that differentiate hyperelliptic functions of arbitrary genus. It provides direct, explicit proofs that the families $\{\mathcal{D}_k\}$ are commuting differential operators and that each $\mathcal{D}_k$ annihilates the polynomials $\lambda_{2j+2}$ defined via generating functions, yielding explicit integrals of the dynamics. The main contributions are: (i) the commutativity of the odd-indexed operators $\mathcal{D}_k$, and (ii) the annihilation $\mathcal{D}_k(\lambda_{2j+2})=0$, linking polynomial invariants to genus-general hyperelliptic function differentiation. These results connect infinite-dimensional polynomial vector-field dynamics with hyperelliptic function theory and the geometry of the universal Jacobian fibrations across 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 $\mathbb{C}^\infty$.