Asymptotic solutions to algebraic equation

We consider a polynomial in three variables near singular point, where the polynomial and its partial derivatives vanish. We propose a method of computation of asymptotic expansions for all branches of the set of roots of the polynomial. We distinguish three types of expansions. The method of computation is based on the spatial Power Geometry. We show an implementation of the method on a polynomial in three variables of the degree six, and we compute asymptotic expansions at infinity and at degenerate singular point of the polynomial.