The problem of differentiation of an Abelian function over its parameters

Theory of Abelian functions was a central topic of the 19th century mathematics. In mid-seventies of the last century a new wave arose of investigation in this field in response to the discovery that Abelian functions provide solutions of a number of challenging problems of modern Theoretical and Mathematical Physics. In a cycle of our joint papers published in 2000–05, we have developed a theory of multivariate sigma-function, an analogue of the classic Weierstrass sigma-function. A sigma-function is defined on a cover of U , where U is the space of a bundle p : U → B defined by a family of plane algebraic curves of fixed genus. The base B of the bundle is the space of the family parameters and a fiber J_b over b ∈ B is the Jacobi variety of the curve with the parameters b. A second logarithmic derivative of the sigma-function along the fiber is an Abelian function on J_b. Thus, one can generate a ring F of fiber-wise Abelian functions on U. The problem to find derivations of the ring F along the base B is a reformulation of the classic problem of differentiation of Abelian functions over parameters. Its solution is relevant to a number of topical applications. This work presents a solution of this problem recently found by the authors. Our method of solution essentially employs the results from Singularity Theory about vector fields tangent to the discriminant of a singularity y^n -x^s, gcd(n, s) = 1