turkmath.org

I will discuss the vast area of research (in which I am not an expert) related to Hilbert's 17th problem, namely, positivity of real polynomials vs. their representation as sums of squares (SOS). As is well known, "most" PSD (positive semi definite) forms in more than two variables are not SOS of polynomials, although they are SOS of rational function. I will consider a few simplest classical counterexamples, and then I will outline the construction part of our recent paper (in collaboration with Erwan Brugallé, Ilia Itenberg, and Frédéric Mangolte). Thinking that we were dealing with Hilbert's 16th problem (widely understood, i.e., topology of real algebraic varieties), we constructed real plane algebraic curves with large finite numbers of real points. These curves provide new lower bounds on the denominators needed to represent a PSD ternary form as a SOS of rational functions.