Rational analogs of abelian functions

Abstract

In the theory of Abelian functions on Jacobians, the key role is played by entire functions that satisfy the Riemann vanishing theorem (see, for instance, [9]). Here we introduce polynomials that satisfy an analog of this theorem and show that these polynomials are completely characterized by this property. By rational aalalogs of Abel ian functions we mean logarithmic derivatives of orders /> 2 of tlmse 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 equat ions [1, 7], special attention was focused on Schur polynomials [5, 6]. Since a Schur polynomial corresponding to all arbitrary partition leads to a rational solution of the Kadomtsev-Petviashvili hierarchy, tile problem of connecting the above solutions with those defined in terms of Abelian functions on Jacobians naturally arose. Our results open the way toward solving this problem on the basis of the Riemann vanishing theorem. We demons t ra t e our approach by the example of Weierstrass sequences defined by a pair of coprime numbers n and s. Each of these sequences generates a class of plane curves of genus g ---(n 1)(s 1)/2 defined by equat ions of the form