Positivity and Sums of Squares

真实的代数几何研究与“真实的世界”的数学建模有关的几何对象，即，物理对象的数学描述，以及自然和人为的过程中的科学和技术。在真实的代数几何中考虑的最基本的函数是具有真实的系数的n元多项式函数。所考虑的功能域是由多项式方程和不等式描述的真实的n-空间的子集。一个基本的问题是确定这样一个函数在这样一个区域上的最小（或最大）值。这是一个很难的问题，一般来说，但各种快速算法的基础上半定规划，最近，几何规划可以用来获得估计的最小值。这些算法的理论基础是正多项式和平方和之间存在的密切关系。正多项式与平方和之间关系的研究是一个古老的课题，可以追溯到世纪后期希尔伯特和赫维茨的工作。里程碑式的成就包括阿廷的解决希尔伯特的第17个问题于1927年，Krivine的Positivstellenkraft于1964年，和Schmuedgen的Positivstellenkraft（处理的情况下，域的问题是紧凑的）于1991年。所提出的研究的主要目的是更好地理解这种关系，并应用这些信息来了解这些快速算法在实际情况下的表现。

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.