Hyperelliptic Sigma Functions and Adler–Moser Polynomials

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.