Abstract Hilbert’s 17th problem asks whether every nonnegative polynomial can be a sum of squares rational functions. It has been answered affirmatively by Artin. However, the question as to given is polynomials still central in real algebraic geometry. In this paper, we solve completely for associated with isoparametric polynomials, initiated E. Cartan, which define focal submanifolds correspo...

This paper is the second in a series of three papers devoted to sums squares and hypoellipticity infinitely degenerate operators. In first we established sharp ω-monotonicity criterion for writing smooth nonnegative function f that flat at, positive away from, origin, as finite sum C2,δ functions some δ>0, namely ω-monotone Hölder modulus continuity ω. Counterexamples were provided any larger c...

We study dimensions of the faces of the cone of nonnegative polynomials and the cone of sums of squares; we show that there are dimensional differences between corresponding faces of these cones. These dimensional gaps occur in all cases where there exist nonnegative polynomials that are not sums of squares. As either the degree or the number of variables grows the gaps become very large, asymp...

Erdős and Rényi proposed in 1960 a probabilistic model for sums of s integral sth powers. Their model leads almost surely to a positive density for sums of s pseudo sth powers, which does not reflect the case of sums of two squares. We refine their model by adding arithmetical considerations and show that our model is in accordance with a zero density for sums of two pseudo-squares and a positi...

Commonly used sums-of-squares-based error or deviation statisticsdlike the standard deviation, the standard error, the coefficient of variation, and the root-mean-square errordoften are misleading indicators of average error or variability. Sums-of-squares-based statistics are functions of at least two dissimilar patterns that occur within data. Both the mean of a set of error or deviation magn...

Turán problems in extremal combinatorics concern asymptotic bounds on the edge densities of graphs and hypergraphs that avoid specified subgraphs. The theory of flag algebras proposed by Razborov provides powerful semidefinite programming based methods to find sums of squares that establish edge density inequalities in Turán problems. Working with polynomial analogs of the flag algebra entities...