Related papers: Gram Spectrahedra

Divided into three parts, the first marks out enormous geometric issues with the notion of quasi-freenss of an algebra and seeks to replace this notion of formal smoothness with an approximation by means of a minimal unital commutative…

Using the theory of signatures of hermitian forms over algebras with involution, developed by us in earlier work, we introduce a notion of positivity for symmetric elements and prove a noncommutative analogue of Artin's solution to…

Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…

Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance…

We introduce positive Gorenstein ideals. These are Gorenstein ideals in the graded ring $\RR[x]$ with socle in degree 2d, which when viewed as a linear functional on $\RR[x]_{2d}$ is nonnegative on squares. Equivalently, positive Gorenstein…

Let $\Gamma$ be a finitely presented group and $G$ a linear algebraic group over $\mathbb{R}$. A representation $\rho:\Gamma\rightarrow G(\mathbb{R})$ can be seen as an $\mathbb{R}$-point of the representation variety $\mathfrak{R}(\Gamma,…

This paper is devoted to the construction of polynomial 2-surfaces which possess a polynomial area element. In particular we study these surfaces in the Euclidean space $\mathbb R^3$ (where they are equivalent to the PN surfaces) and in the…

We introduce a method for proving Sum-of-Squares (SoS)/ Lasserre hierarchy lower bounds when the initial problem formulation exhibits a high degree of symmetry. Our main technical theorem allows us to reduce the study of the positive…

In this paper we explore inequalities between symmetric homogeneous polynomials of degree four of three real variables and three nonnegative real variables. The main theorems describe the cases in which the smallest possible coefficient is…

Given a linear map on the vector space of symmetric matrices, every fiber intersected with the set of positive semidefinite matrices is a spectrahedron. Using the notion of the fiber body we can build the average over all such fibers and…

Writing the values of Krawtchouk polynomials as matrices, we consider weighted partial sums along columns. For the general case, we find an identity that, in the symmetric case yields a formula for such partial sums. Complete sums of…

Many problems of systems control theory boil down to solving polynomial equations, polynomial inequalities or polyomial differential equations. Recent advances in convex optimization and real algebraic geometry can be combined to generate…

We study the differential and metric structures of the set of real square roots of a non-singular real matrix, under the assumption that the matrix and its square roots are semi-simple, or symmetric, or orthogonal.

The relationship between nonnegative polynomials and sums of squares is a classical topic in real algebraic geometry. We study \emph{stubborn polynomials} $f$ on a real variety $X$, which are polynomials nonnegative on $X$, such that no odd…

In this note, simple proofs of certain well-known results involving the positive square root of positive matrices are given.

For a fixed quadratic polynomial $\mathfrak{p}$ in $n$ non-commuting variables, and $n$ independent $N\times N$ complex Ginibre matrices $X_1^N,\dots, X_n^N$, we establish the convergence of the empirical spectral distribution of $P^N…

Polynomial reduction is one of the main tools in computational algebra with innumerable applications in many areas, both pure and applied. Since many years both the theory and an efficient design of the related algorithm have been solidly…

We realize many sharp spectral bounds of the spectral radius of a nonnegative square matrix $C$ by using the largest real eigenvalues of suitable matrices of smaller sizes related to $C$ that are very easy to find. As applications, we give…

Knowing the symmetries of a polyhedron can be very useful for the analysis of its structure as well as for practical polyhedral computations. In this note, we study symmetry groups preserving the linear, projective and combinatorial…

We determine the multiplicities of irreducible summands in the symmetric and the exterior squares of hook representations of symmetric groups over an algebraically closed field of characteristic zero.