Hyperelliptic sigma functions and AdlerMoser polynomials
Abstract
In a 2004 paper by V. M. Buchstaber and D. V. Leykin, published in "Functional Analysis and Its Applications," for each $g > 0$, a system of $2g$ multidimensional heat equations in a nonholonomic frame was constructed. The sigma function of the universal hyperelliptic curve of genus $g$ is a solution of this system. In the work arXiv:2007.08966 explicit expressions for the Schrödinger operators that define the equations of the system considered were obtained in the hyperelliptic case. In this work we use these results to show that if the initial condition of the system considered is polynomial, then the solution of the system is uniquely determined up to a constant factor. This has important applications in the wellknown problem of series expansion for the hyperelliptic sigma function. We give an explicit description of the connection of such solutions to wellknown BurchnallChaundy polynomials and AdlerMoser polynomials. We find a system of linear secondorder differential equations that determines the corresponding AdlerMoser polynomial.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.10896
 Bibcode:
 2021arXiv210610896B
 Keywords:

 Mathematical Physics;
 Mathematics  Complex Variables;
 Mathematics  Dynamical Systems;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems