Our first contribution in this paper is to prove that three natural sum of squares (sos) based
sufficient conditions for convexity of polynomials, via the definition of convexity, its first order …

Convex sets arising in a variety of applications are well-defined for every relevant
dimension. Examples include the simplex and the spectraplex that correspond, respectively …

We study the geometry of the image of the nonnegative orthant under the power-sum map
and the elementary symmetric polynomials map. After analyzing the image in finitely many …

The arithmetic mean/geometric mean inequality (AM/GM inequality) facilitates classes of
nonnegativity certificates and of relaxation techniques for polynomials and, more generally …

We prove that convex ternary quartic forms are sum-of-squares-convex (sos-convex). This
result is in a meaningful sense the``convex analogue''a celebrated theorem of Hilbert from …

An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate
general symmetric ideals to the so called Specht ideals generated by all Specht polynomials …

We consider cones of real forms which are sums of squares and invariant under a (finite)
reflection group. Using the representation theory of these groups we are able to use the …

A set of $ k $ orthonormal bases of $\mathbb C^ d $ is called mutually unbiased if $|\langle
e, f\rangle|^ 2= 1/d $ whenever $ e $ and $ f $ are basis vectors in distinct bases. A natural …

We study the limits of the cones of symmetric nonnegative polynomials and symmetric sums
of squares, when expressed in power-mean or monomial-mean basis. These limits …

We study how well functions over the boolean hypercube of the form $ f_k (x)=(| x|-k)(| x|-k-1)
$ can be approximated by sums of squares of low-degree polynomials, obtaining good …