Writing Positive Polynomials as Sums of (Few) Squares

[摘要] In 1927, Artin proved that a real polynomial that is positivesemidefinite is a sum of squares of rational functions, thussolving Hilbert’s 17th problem. We review Artin’s Theoremand its posterity, browsing through basic examples, classicalresults and recent developments. We focus on a question firstconsidered by Pfister: can one write a positive semidefinitepolynomial as a sum of few squares.