Algorithms for Solving an Algebraic Equation

For finding global approximate solutions to an algebraic equation in n unknowns, the Hadamard open polygon for the case n = 1 and Hadamard polyhedron for the case n = 2 are used. The solutions thus found are transformed to the coordinate space by a translation (for n = 1) and by a change of coordinates that uses the curve uniformization (for n = 2). Next, algorithms for the local solution of the algebraic equation in the vicinity of its singular (critical) point for obtaining asymptotic expansions of one-dimensional and two-dimensional branches are presented for n = 2 and n = 3. Using the Newton polygon (for n = 2), the Newton polyhedron (for n = 3), and power transformations, this problem is reduced to situations similar to those occurring in the implicit function theorem. In particular, the local analysis of solutions to the equation in three unknowns leads to the uniformization problem of a plane curve and its transformation to the coordinate axis. Then, an asymptotic expansion of a part of the surface under examination can be obtained in the vicinity of this axis. Examples of such calculations are presented.