Explicit formulas for differentiation of hyperelliptic functions, multidimensional sigma functions and applications

In the 1882 paper by F. G. Frobenius and L. Stickelberger [1], the problems of differentiation of elliptic functions over parameters of elliptic curves and over periods of elliptic functions were solved. In the talk we will consider a modern multidimensional generalization of these problems and present their explicit solutions.
Our approach is based on methods from [2] and our results in the theory of multidimensional sigma functions defined for families of hyperelliptic curves. Using these results, we determine such sigma functions as solutions of explicitly defined systems of equations. The sigma functions are used to construct the field of hyperelliptic functions, the multidimensional generalization of the field of elliptic functions. Hyperelliptic functions are multidimensional meromorphic functions with sets of periods determined by hyperelliptic curves.
The 1974 paper by S. P. Novikov [3] formed the basis for the development of widely known algebraic-geometric methods in soliton theory and mathematical physics. The methods allow to obtain solutions of fundamental dynamical systems in terms of multidimensional meromorphic functions with sets of periods determined by algebraic curves. We will show applications of our results to such solutions.
Our main results are presented in [4].