Positive polynomials and sums of squares

真正的代数几何研究与“真实世界”的数学建模有关的几何对象，即物理对象以及科学技术中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数系数的n个实数变量的多项式函数。所考虑的域是由多项式方程和不等式定义的实n空间的子集。一个基本问题是确定这样一个函数在这样一个域上的最小值（或最大值）。这通常是一个难题，但各种基于半定规划的快速算法可用于获得最小值的估计。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世纪末希尔伯特的工作。里程碑包括1927年Artin对Hilbert第17问题的解决，1964年Krivine的Positivstellensatz，以及1991年Schmuedgen的Positivstellensatz（处理所讨论的域是紧化的情况）。本研究的主要目的是更好地理解这种关系，并应用这些信息来更好地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 description 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 real variables with real coefficients. The domains considered are subsets of real n-space defined 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 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 work of Hilbert 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 when 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 better how these fast algorithms will perform in practical situations.