
Author(s) 
Gindikin, S. Volevich, L.R. 

Title  The Method of Newton's Polyhedron in the Theory of Partial Differential Equations 
Publisher  Kluwer Academic Publishers 
Year of publication  1992 
Reviewed by  VALERII V. TROFIMOV 
The book under review elucidates the method of Newton's polyhedron for solving some problems in the theory of partial differential equations. Let us recall, the Newton's polyhedron of a polynomial in several variables is the convex hull of the set of exponents of its monomials. The three problems in the theory of partial differential equations are analysed from the Newton's polyhedron stand point: i) investigation of a special class of hypoelliptic operators; ii) obtaining of the energy estimates in Cauchy's problem; iii) discussion of the generalized operators of principal type.
The roughs contents of the book are as follows, 1. Twosided estimates for polynomials related to Newton's polygon and their application to studying local properties of partial differential operators in two variables; 2. Parabolic operators associated with Newton's polygon; 3. Dominantly correct operators; 4. Operators of principal type associated with Newton's polygon; 5. Twosided estimates in several variables relating to Newton's polyhedra; 6. 0perators of principal type associated with Newton's polyhedron; 7. The method of energy estimates in Cauchy's problem.
At the present time there are not the books devoted specially to harnessing of the Newton's polyhedrons. The level of the book under review is postgraduated. The significant algebraic technique applicable to many other problems is enclosed.
The Method of Newton's Polyhedron in the Theory of Partial Differential Equations shall be of interest to researchers and graduate students. I can recommend the students, experts, and libraries alike purchase of this book.