Uspekhi Mat. Nauk, 2021,	Volume 76,	Issue 4(460),	Pages 37–104	(Mi umn10007)
Integrable polynomial Hamiltonian systems and symmetric powers of plane algebraic curves
V. M. Buchstabera, A. V. Mikhailovbc
a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b University of Leeds, Leeds, UK
c Centre of Integrable Systems, P.G. Demidov Yaroslavl State University
Abstract: This survey is devoted to integrable polynomial Hamiltonian systems associated with symmetric powers of plane algebraic curves. We focus our attention on the relations (discovered by the authors) between the Stäckel systems, Novikov's equations for the $g$th stationary Korteweg–de Vries hierarchy, the Dubrovin–Novikov coordinates on the universal bundle of Jacobians of hyperelliptic curves, and new systems obtained by considering the symmetric powers of curves when the power is not equal to the genus of the curve.
Bibliography: 52 titles.
Keywords: polynomial Hamiltonian systems, Stäckel systems, Korteweg–de Vries hierarchy, symmetric powers of curves, Abelian functions, systems of hydrodynamical type.
Funding Agency Grant Number
Ministry of Science and Higher Education of the Russian Federation 075-02-2021-1397
EPSRC EP/V050451/1
The research of the second author was carried out with the support of the Ministry of Science and Higher Education of the Russian Federation (contract no. 075-02-2021-1397) and the Engineering and Physical Sciences Research Council (grant no. EP/V050451/1).
DOI: https://doi.org/10.4213/rm10007
Full text: PDF file (957 kB)
First page: PDF file
Russian Mathematical Surveys, 2021, 76:4, 587–652
UDC: 517.938+512.77
MSC: 14H70, 14K25, 37J35
Citation: V. M. Buchstaber, A. V. Mikhailov, "Integrable polynomial Hamiltonian systems and symmetric powers of plane algebraic curves", Uspekhi Mat. Nauk, 76:4(460) (2021), 37–104; Russian Math. Surveys, 76:4 (2021), 587–652
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http://mi.mathnet.ru/eng/umn10007
https://doi.org/10.4213/rm10007
http://mi.mathnet.ru/eng/umn/v76/i4/p37
First page: 14