Positivity and Sums of Squares

实代数几何研究与“真实世界”的数学建模有关的几何对象，即对物理对象以及科学技术中的自然和人为过程的数学描述。实代数几何中最基本的函数是具有实系数的n元多项式函数。所考虑的函数的区域是实n-空间的子集，用多项式方程和不等式来描述。一个基本问题是确定这样一个域上这样一个函数的最小值(或最大值)。这通常是一个困难的问题，但基于半定规划和最近的几何规划的各种快速算法可以用来获得最小值的估计。这些算法的理论基础是正多项式与平方和之间存在的密切关系。研究正多项式与平方和之间的关系是一个古老的课题，可以追溯到19世纪末Hilbert和Hurwitz的工作。里程碑包括Artin在1927年解决了Hilbert的第17个问题，Krivine在1964年解决了Positivstellensatz，Schmuedgen在1991年解决了领域是紧凑的情况。建议研究的主要目的是更好地理解这种关系，并应用这些信息来理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.