Positivity and Sums of Squares

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Real algebraic geometry studies geometrical objects arising in connection with the mathematical modeling of the "real world", i.e., the mathematical descriptions of physical objects as well as of natural and man-made processes in science and technology. The most basic functions considered in real algebraic geometry are the polynomial functions in n variables with real coefficients. The domains of the functions considered are subsets of real n-space which are described by polynomial equations and inequalities. A basic problem is to determine the minimum (or maximum) value of such a function on such a domain. This is a hard problem in general, but various fast algorithms based on semidefinite programming and, more recently, on geometric programming can be used to obtain estimates for the minimum value. The theoretical basis for these algorithms is the close relationship that exists between positive polynomials and sums of squares. The study of the relationship between positive polynomials and sums of squares is an old subject, dating back to the work of Hilbert and Hurwitz in the late 19th century. Landmarks include Artin's solution of Hilbert's 17th problem in 1927, Krivine's Positivstellensatz in 1964, and Schmuedgen's Positivstellensatz (dealing with the case where the domain in question is compact) in 1991. The main object of the proposed research is to understand this relationship better, and apply this information to understand how these fast algorithms will perform in practical situations.