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This article is cited in 2 scientific papers (total in 2 papers)
Hyperelliptic Sigma Functions and Adler–Moser Polynomials
V. M. Buchstaber, E. Yu. Bunkova Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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.
Keywords:
Schrödinger operator, polynomial Lie algebra, polynomial dynamical system, heat equation in a
nonholonomic frame, differentiation of Abelian functions with respect to parameters, Adler–Moser
polynomial, Burchnall–Chaundy equation, Korteweg–de Vries equation.
Received: 18.06.2021 Revised: 18.06.2021 Accepted: 21.06.2021
Citation:
V. M. Buchstaber, E. Yu. Bunkova, “Hyperelliptic Sigma Functions and Adler–Moser Polynomials”, Funktsional. Anal. i Prilozhen., 55:3 (2021), 3–25; Funct. Anal. Appl., 55:3 (2021), 179–197
Linking options:
https://www.mathnet.ru/eng/faa3915https://doi.org/10.4213/faa3915 https://www.mathnet.ru/eng/faa/v55/i3/p3
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Abstract page: | 381 | Full-text PDF : | 137 | References: | 60 | First page: | 30 |
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