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Rational analogues of Abelian functions. (English. Russian original) Zbl 1056.14049
Funct. Anal. Appl. 33, No. 2, 83-94 (1999); translation from Funkts. Anal. Prilozh. 33, No. 2, 1-15 (1999).
From the introduction: In the theory of abelian functions on Jacobians, the key role is played by entire functions that satisfy the Riemann vanishing theorem. Here we introduce polynomials that satisfy an analog of this theorem and show that these polynomials are completely characterized by this property. By rational analogs of abelian functions we mean logarithmic derivatives of orders \(\geq 2\) of these polynomials. We call the polynomials thus obtained the Schur-Weierstrass polynomials because they are constructed from classical Schur polynomials, which, however, correspond to special partitions related to Weierstrass sequences. Recently, in connection with the problem of constructing rational solutions of nonlinear integrable equations [M. Adler and J. Moser, Commun. Math. Phys. 61, 1–30 (1978; Zbl 0428.35067) and I. M. Krichever, Zap. Nauchn. Sem. LOMI, 84, No. 1, 117–130 (1979; Zbl 0413.35008)], special attention was focused on Schur polynomials [J. J. Duisterman and F. A. Gruenbaum, Commun. Math. Phys. 103, 177–204 (1986; Zbl 0625.34007) and V. G. Kac, Infinite Dimensional Lie Algebras, Birkhäuser (1983; Zbl 0584.17007)]. Since a Schur polynomial corresponding to an arbitrary partition leads to a rational solution of the Kadomtsev-Petviashvili hierarchy, the problem of connecting the above solutions with those defined in terms of abelian functions on Jacobians naturally arose. One results open the way toward solving this problem on the basis of the Riemann vanishing theorem.

MSC:
14H70 Relationships between algebraic curves and integrable systems
14H40 Jacobians, Prym varieties
14H55 Riemann surfaces; Weierstrass points; gap sequences
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
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References:
[1] M. Adler and J. Moser, ”On a class of polynomials connected with the Korteweg-de Vries equation,” Commun. Math. Phys.,61, 1–30 (1978). · Zbl 0428.35067 · doi:10.1007/BF01609465
[2] V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, ”Kleinian functions, hyperelliptic Jacobians and applications,” In: Reviews in Mathematics and Mathematical Physics (S. P. Novikov and I. M. Krichever, eds.), Vol. 10, No. 2, Gordon & Breach, London, 1997, pp. 1–125. · Zbl 0911.14019
[3] V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, ”\(\sigma\)-Functions of (n,s)-curves,” Usp. Mat. Nauk,54, No. 3 (1999). · Zbl 0911.14019
[4] V. M. Buchstaber and E. G. Rees, ”Multivalued groups,n-Hopf algebras andn-ring homorphisms,” In: Lie Groups and Lie Algebras, Kluwer, 1998, pp. 85–107. · Zbl 0903.20039
[5] J. J. Duistermaat and F. A. Gruenbaum, ”Differential equations in spectral parameters,” Commun. Math. Phys.,103, 177–204 (1986). · Zbl 0625.34007 · doi:10.1007/BF01206937
[6] V. G. Kac, Infinite Dimensional Lie Algebras, Birkhäuser, 1983. · Zbl 0537.17001
[7] I. M. Krichever, ”On the rational solutions of the Zaharov-Shabat equations and completely integrable systems ofN particles on a line,” Zap. Nauchn. Sem. LOMI,84, No. 1, 117–130 (1979). · Zbl 0413.35008
[8] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1995. · Zbl 0824.05059
[9] D. Mumford, Tata Lectures on Theta, I, II, Birkhäuser, 1983, 1984.
[10] G. Polya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, V. 2, Springer-Verlag, 1964.
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