Inversion of many-dimensional Mellin transforms and solutions of algebraic equations

For an arbitrary pair of convex domains U, Theta subset of R-n one introduces mirror-symmetric vector spaces M-theta(U) and W-U(theta) consisting of holomorphic functions in the corresponding domains and taken to each other by the direct and the inverse Mellin transformations. As applications, a generalization of the classical integral Mellin transform for a solution y(X) of the general algebraic equation is obtained and the convergence domain of the Mellin-Barnes hypergeometric integral representing the solution y(X) is found.