Related papers: Noncommutative polynomials nonnegative on a variet…
Given a monic linear pencil L in g variables let D_L be its positivity domain, i.e., the set of all g-tuples X of symmetric matrices of all sizes making L(X) positive semidefinite. Because L is a monic linear pencil, D_L is convex with…
The free closed semialgebraic set $D_f$ determined by a hermitian noncommutative polynomial $f$ is the closure of the connected component of $\{(X,X^*)\mid f(X,X^*)>0\}$ containing the origin. When $L$ is a hermitian monic linear pencil,…
This article studies algebraic certificates of positivity for noncommutative (nc) operator-valued polynomials on matrix convex sets, such as the solution set $D_L$, called a free Hilbert spectrahedron, of the linear operator inequality…
We derive some Positivstellensatz\"e for noncommutative rational expressions from the Positivstellensatz\"e for noncommutative polynomials. Specifically, we show that if a noncommutative rational expression is positive on a polynomially…
The (matricial) solution set of a Linear Matrix Inequality (LMI) is a convex basic non-commutative semi-algebraic set. The main theorem of this paper is a converse, a result which has implications for both semidefinite programming and…
Hermitian linear matrix pencils are ubiquitous in control theory, operator systems, semidefinite optimization, and real algebraic geometry. This survey reviews the fundamental features of the matricial solution set of a linear matrix…
This paper establishes new Positivstellens\"atze for polynomials that are positive on sets defined by polynomial matrix inequalities (PMIs). We extend the classical Handelman and Krivine-Stengle theorems from the scalar inequality setting…
A subspace of an algebra with involution is called a Lie skew-ideal if it is closed under Lie products with skew-symmetric elements. Lie skew-ideals are classified in central simple algebras with involution (there are eight of them for…
Positivstellensatz is a fundamental result in real algebraic geometry providing algebraic certificates for positivity of polynomials on semialgebraic sets. In this article Positivstellens\"atze for trace polynomials positive on…
Suppose p is a symmetric matrix whose entries are polynomials in freely noncommutating variables and p(0) is positive definite. Let D(p) denote the component of zero of the set of those g-tuples X of symmetric matrices (of the same size)…
Given a polynomial $x \in {\mathbb R}^n \mapsto p(x)$ in $n=2$ variables, a symbolic-numerical algorithm is first described for detecting whether the connected component of the plane sublevel set ${\mathcal P} = \{x : p(x) \geq 0\}$…
We prove a Positivstellensatz for operator-valued noncommutative polynomials that are positive on matrix convex sets. Specifically, let $p$ be an operator-valued polynomial in $B(H)\otimes C<x>$ of degree at most $2d+1$, where $H$ is…
A linear polyomial non-negative on the non-negativity domain of finitely many linear polynomials can be expressed as their non-negative linear combination. Recently, under several additional assumptions, Helton, Klep, and McCullough…
We propose and discuss how basic notions (quadratic modules, positive elements, semialgebraic sets, Archimedean orderings) and results (Positivstellensaetze) from real algebraic geometry can be generalized to noncommutative $*$-algebras. A…
Hilbert's Nullstellensatz characterizes polynomials that vanish on the vanishing set of an ideal in C[x]. In the free algebra C<X> the vanishing set of a two-sided ideal I is defined in a dimension-free way using images in…
The question how to certify non-negativity of a polynomial function lies at the heart of Real Algebra and also has important applications to Optimization. In this article we investigate the question of non-negativity in the context of…
The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov have proved that any real zero polynomial in two variables has a determinantal…
This article extends the classical Real Nullstellensatz to matrices of polynomials in a free $\ast$-algebra $\RR\axs$ with $x=(x_1, \ldots, x_n)$. This result is a generalization of a result of Cimpri\vc, Helton, McCullough, and the author.…
There has recently been ample interest in the question of which sets can be represented by linear matrix inequalities (LMIs). A necessary condition is that the set is rigidly convex, and it has been conjectured that rigid convexity is also…
Hilbert's Nullstellensatz is one of the most fundamental correspondences between algebra and geometry, and has inspired a plethora of noncommutative analogs. In last two decades, there has been an increased interest in understanding…