The Method of Newton’s Polyhedron in the Theory of Partial Differential Equations

Newton's polyhedron of a polynomial in several variables is the convex hull of the set of exponents of its monomials completed in some way. The boundary of Newton's polyhedron can be interpreted as one of the possible generalizations of the degree of a polynomial in one variable to the case of several variables. This is a much more informative notion than that of the ordinary degree. At the same time, it is not invariant with respect to linear transformations of coordinates and thus is related to a fixed coordinate system. An intermediate notion between the degree and Newton's polyhedron is the notion of a weighed degree in which the calculation of the "degree" of a monomial is performed by means of summation of the degrees of the various variables with different weights. Newton's polyhedron accunmlates information about the various weighted degrees and principal parts which correspond to its faces.
Recently it has been revealed that Newton's polyhedron is a suitable technical means in extremely versatile mathematical problems. In thi8 book we develop the method of Newton's polyhedron for some problems in t h ( ~ theory of partial differential equations. It splits into two parts, Chapters 1 to 4 and Chapters 5 to 7, where Newton's polygon and Newton's polyhedron are considered. The case of polygon not only makes it possible to consider general constructions in the simpler two-dimensional case but also has some natural multidimensional applications.