Abstract. A telescopic curve is a certain algebraic curve defined by m − 1 equations in the affine space of dimension m, which can be a hyperelliptic curve and an (n, s) curve as a special case. We extend the addition formulae for sigma functions of (n, s) curves to those of telescopic curves. The expression of the prime form in terms of the derivative of the sigma function is also given.
The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus 3 is described in terms of the gradient of it's sigma function. Solutions of corresponding families of polynomial dynamical systems in C 4 with two polynomial integrals are constructed as an application. These systems were introduced in the work of V. M. Buchstaber and A. V. Mikhailov on the basis of commuting vector fields on the symmetric square of algebraic curves. * This is the accepted version. The final publication is available at
Abstract. For a hyperelliptic curve of genus g, it is well known that the symmetric products of g points on the curve are expressed in terms of their Abel-Jacobi image by the hyperelliptic sigma function (Jacobi inversion formulae). Matsutani and Previato gave a natural generalization of the formulae to the more general algebraic curves defined by y r = f (x), which are special cases of (n, s) curves, and derived new vanishing properties of the sigma function of the curves y r = f (x). In this paper we extend the formulae to the telescopic curves proposed by Miura and derive new vanishing properties of the sigma function of telescopic curves. The telescopic curves contain the (n, s) curves as special cases.
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