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A recursive family of differential polynomials generated by the Sylvester identity, and addition theorems for hyperelliptic Kleinian functions. (English. Russian original) Zbl 1126.33301
Funct. Anal. Appl. 31, No. 4, 240-251 (1997); translation from Funkts. Anal. Prilozh. 31, No. 4, 19-33 (1997).
From the introduction: In the present paper we introduce a recursive family of functions $$\{F_n\}){n=1,2,\dots}$$ generated by the relations $F_{n+1} F_{n-1}= \lambda(F_n\partial_{xy} F_n- \partial_x F_n- \partial_x F_n\partial_y F_n)+\mu f(x, y)F^2_n\tag{1}$ with initial conditions $$F_0= 1$$ and $$F_1= f(x,y)$$. It is shown that, for any $$n$$, the function $$F_n(x, y)$$ is a polynomial with integer coefficients in a function $$f(x, y)$$, its partial derivatives, and the parameters $$\lambda$$ and $$\mu$$. The differential polynomials $$F_n(x, y)= D_n(f(x, y))= D_n(f(x, y);\lambda,\mu)$$ are representable as leading principal minors of a matrix, and the family $$\{D_n(f)\}$$ itself is generated by the Sylvester identity for compound determinants.
We apply this recursion to the classical problem of generalizing the relation ${\sigma(u+ v)\sigma(u- v)\over \sigma(u)^2\sigma(v)^2}= \wp(v)- \wp(u)$ for the elliptic Weierstrass functions which plays a key role in the theory and applications of elliptic functions (here the genus is $$g= 1$$), to the case of hyperelliptic Kleinian $$\sigma$$-functions (with genus $$g> 1$$).
A hyperelliptic $$\sigma$$-function of genus $$g$$ is defined as an element of the ring of Riemann $$\theta$$-functions that is automorphic with respect to the action of the modular group $$\text{Sp}(2g,\mathbb{Z})$$. The logarithmic derivatives of the $$\sigma$$-function, $\wp_{ij}=- {\partial^2\over\partial u_i\partial u_j}\ln \sigma(u),\;\wp_{i,j,k}=-{\partial^2\over\partial u_i\partial u_j\partial u_k}\ln\sigma(u),\;i,j,k= 1,\dots, g,$ and so on, are hyperelliptic Abelian functions, i.e., $$2g$$-periodic meromorphic functions.
If the genus $$g$$ is one, then the field of elliptic functions is generated by Weierstrass elliptic functions $$\wp$$ and $$\wp'$$, which uniformize an elliptic curve. In the case $$g> 1$$, the hyperelliptic functions $$\wp_{ij}$$ and $$\wp_{ijk}$$ uniformize the Jacobian variety $$\text{Jac}(V)$$ of a hyperelliptic curve $$V$$, while the even functions $$\wp_{ij}$$ themselves uniformize the Kummer variety $$\text{Kum}(V)= \text{Jac}(V)/(u\to -u)$$, defined as the quotient of the Jacobian with respect to the hyperelliptic involution.
The following assertion is one of the main results of the paper.
For a hyperelliptic Kleinian $$\sigma$$-function of genus $$g\geq 1$$, the following relation holds: ${\sigma(u+ v)\sigma(u- v)\over \sigma(u)^2\sigma(v)^2}= D_g\Biggl(\wp_{gg}(v)- \wp_{gg}(u);\,{1\over 4},{1\over 2}\Biggr),$ where $$\wp_{gg}(v)- \wp_{gg}(u)$$ is regarded as a function of $$x= u_g+ v_g$$ and $$y= u_g- v_g$$, and $$D_g(\;;\;)$$ is the differential polynomial defined by the recursion (1).

##### MSC:
 33E05 Elliptic functions and integrals 14K20 Analytic theory of abelian varieties; abelian integrals and differentials 14H42 Theta functions and curves; Schottky problem 35Q53 KdV equations (Korteweg-de Vries equations) 14H40 Jacobians, Prym varieties 14K25 Theta functions and abelian varieties 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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##### References:
  H. F. Baker, ”On the hyperelliptic sigma functions,” Am. J. Math.,20, 301–384 (1898). · JFM 29.0394.03  L. Königsberger, ”Über die Transformation der Abelshen Funktionen erster Ordnung,” J. Reine Angew. Math.,64 (27), 17–42 (1865). · ERAM 064.1660cj  V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, ”Matrix realization of a hyperelliptic Jacobi variety,” Usp. Mat. Nauk,51, No. 2, 147–148 (1996).  V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, ”Hyperelliptic Kleinian functions and applications,” in: Solitons Geometry and Topology: On the Crossroad, Advances in Math. Sciences, Am. Math. Soc. Transl., Ser. 2, Vol. 179, 1997, pp. 1–34. · Zbl 0911.14019  V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, ”Kleinian functions, hyperelliptic Jacobians and applications,” in: Reviews in Mathematics and Mathematical Physics, Vol. 10, No. 2, Gordon and Breach, London, 1997, pp. 1–125. · Zbl 0911.14019  C. L. Siegel, ”Meromorphe Funktionen auf kompacten analytishen Mannigfaltigkeinten,” Göttingen Nachrichten, 71–77, 1955, Werke, III, 216–222. · Zbl 0064.08201  S. Bochner, ”On the addition theorem for multiply periodic functions,” Proc. Am. Math. Soc.,3, 99–106 (1952). · Zbl 0046.31001  V. V. Golubev, Lectures on the Analytic Theory of Differential Equations [in Russian], GITTL, Moscow-Leningrad, 1950. · Zbl 0038.24201  V. M. Buchstaber and V. Z. Enolskii, ”Abelian Bloch solutions of the two-dimensional Schrödinger equation,” Usp. Mat. Nauk,50, No. 1, 191–192 (1995).  F. W. Nijhoff and V. Z. Enolskii, ”Integrable mappings of KdV type and hyperelliptic addition theorems,” in: Proceedings of the 2nd International Conference on Symmetries and Integrability of Difference Equations (SIDE II), Canterbery, July 1996, Springer, 1997, pp. 1–12. · Zbl 0928.37015  D. V. Leykin, ”On the Weierstrass cubic for hyperelliptic functions,” Usp. Mat. Nauk,50, No. 6, 191–192 (1995).  V. M. Buchstaber and V. Z. Enolskii, ”Explicit description of the hyperelliptic Jacobian on the base of Kleinian {$$\sigma$$}-functions,” Funkts. Anal. Prilozhen.,30, No. 1, 57–60 (1996).  H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1955.  V. M. Buchstaber and E. G. Rees, ”Frobeniusk-characters andn-ring homomorphisms,” Usp. Mat. Nauk,52, No. 2, 159–160 (1997).
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