On Multivariate Hermite- Pade Approximations

The problem of approximating a real-valued locally analytic multivariate function by a multivariate algebraic function, i.e. Hermite-Pade approximation is considered. The existence theorem is obtained. The basic properties used to define the rational approximants can be preserved almost intact. Especially, the local behavior of the diagonal bivariate quadratic algebraic function approximation is analyzed