Let f(x1,x2, ..., xn) be a polynomial function in n variables with real coefficients. If f can be expressed as the sum of squares of rational functions then it is trivial to see that f is always nonnegative. But is the converse statement true?
Whether any function which takes only nonnegative values can be written as a sum of squares of rational functions was the content of Hilbert's 17th problem. If f is a function of a single variable then the answer is yes, and in fact f can be expressed as the sum of squares of polynomials. If n = 2, Hilbert himself had already proved that there were positive functions which could not be expressed as the sum of squares of polynomials (one example is f(x,y)= x4y2+ x2y4– 3x2y2+ 1), but that they can all be expressed as the sum of squares of rational functions.
In 1927, Emil Artin was able to prove that Hilbert's 17th problem is true for all n, and actually was able to use model theory to generalize the problem to arbitrary real closed fields. However, like most good problems in mathematics the solution to Hilbert's 17th problem was not the end of the story, as this problem and the work done to solve it laid the groundwork for the field of real algebraic geometry, also known as semialgebraic geometry. This area looks at subsets of Rn which are defined by polynomial equations and inequalities and shares some techniques with classical (complex) algebraic geometry, but has many important differences as well.
Murray Marshall's new book Positive Polynomials and Sums of Squares begins with Hilbert's 17th problem and related work, and quickly takes the reader on a tour of real algebraic geometry and many of the results in this area over the last century. The last two decades have seen many advances in this work, much of which has been inspired by new connections between real algebraic geometry and the moment problem, which asks when a given linear map corresponds to a Borel measure on a given closed subset of n-dimensional space. Recent work by Schmudgen, Jacobi, and others have connected these areas and Marshall takes on the difficult task of giving the relevant background necessary for a beginning graduate student to understand the different questions and theorems involved.
Later chapters in the book get quite technical, as Marshall consider the connections between real algebraic geometry and quadratic forms, semidefinite programming, and optimization, as well as algorithmic questions that are needed to make many of the results constructive.
Darren Glass (email@example.com) is an Assistant Professor at Gettysburg College.