Hilbert’s 17th Problem: Analyzing Vanishing Orders Via Ordering

(Abstract)

Hilbert’s 17^th problem is to decompose a nonnegative real polynomial (in several variables) as the sum of squares of rational functions. If the polynomial is strictly positive, explicit computation by Polya and Reznick provide constructive methods for such decomposition. However, if the polynomial is nonnegative, the zero set presents a geometric obstruction to such constructive methods. Artin analyzed these zeros by imposing an optimal ordering using the theory of real closed field, providing an affirmative, yet non-constructive, answer to Hilbert’s problem.

In this talk, we discuss other analogues of Hilbert’s problem to illustrate that the zero sets present the main difficulty. For example, Handelman and Mok-To considered the problem of decomposing a polynomial nonnegative on the simplex. Varolin and Catlin-D’Angelo considered the problem of decomposing bihomogeneous polynomials nonnegative on complex Euclidean space.