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 our previous work, published in “Functional Analysis and Its Applications,” explicit expressions for the Schrödinger operators that define the equations of this system were obtained in the hyperelliptic case.

In this work we use these results to show that if the initial condition of the system is polynomial, then its solution is uniquely determined up to a constant factor. This has important applications in the well-known problem of series expansion for the hyperelliptic sigma function. We give an explicit description of the connection between such solutions and the well-known Burchnall–Chaundy polynomials and Adler–Moser polynomials. We find a system of linear second-order differential equations that determines the corresponding Adler–Moser polynomial.

中文翻译：

---

超椭圆 Sigma 函数和 Adler-Moser 多项式

摘要

在 VM Buchstaber 和 DV Leykin 于 2004 年发表在“Functional Analysis and Its Applications”上的一篇论文中，对于每个\(g > 0\)，构建了一个非完整框架中的\(2g\)多维热方程系统。属\(g\)的通用超椭圆曲线的 sigma 函数是该系统的一个解。在我们之前发表在“函数分析及其应用”上的工作中，在超椭圆情况下获得了定义该系统方程的薛定谔算子的显式表达式。

在这项工作中，我们使用这些结果来表明，如果系统的初始条件是多项式的，那么它的解是唯一确定的，直到一个常数因子。这在众所周知的超椭圆 sigma 函数的级数展开问题中具有重要应用。我们明确描述了这些解与著名的 Burchnall-Chaundy 多项式和 Adler-Moser 多项式之间的联系。我们找到了一个线性二阶微分方程系统，它确定了相应的 Adler-Moser 多项式。