Related papers: Noncommutative polynomials nonnegative on a variet…

Let $V$ be a real algebraic variety with singularities and $f$ be a real polynomial non-negative on $V$. Assume that the regular locus of $V$ is dense in $V$ by the usual topology. Using Hironaka's resolution of singularities and…

Let L be the zero set of a nonconstant monic polynomial with complex coefficients. In the context of constructive mathematics without countable choice, it may not be possible to construct an element of L. In this paper we introduce a notion…

This article resides in the realm of the noncommutative (free) analog of real algebraic geometry - the study of polynomial inequalities and equations over the real numbers - with a focus on matrix convex sets $C$ and their projections $\hat…

Motivated by classical notions of partial convexity, biconvexity, and bilinear matrix inequalities, we investigate the theory of free sets that are defined by (low degree) noncommutative matrix polynomials with constrained terms. Given a…

Let a and x denote tuples of (jointly) freely noncommuting variables. A square matrix valued polynomial p in these variables is naturally evaluated at a tuple (A,X) of symmetric matrices with the result p(A,X) a square matrix. The…

Positivstellens{\"a}tze are a group of theorems on the positivity of involution algebras over $\mathbb{R}$ or $\mathbb{C}$. One of the most well-known Positivstellensatz is the solution to Hilbert's 17th problem given by E. Artin, which…

The question of how to certify the non-negativity of a polynomial function lies at the heart of Real Algebra and it also has important applications to Optimization. In the setting of symmetric polynomials Timofte provided a useful way of…

A noncommutative (nc) polynomial is called (globally) trace-positive if its evaluation at any tuple of operators in a tracial von Neumann algebra has nonnegative trace. Such polynomials emerge as trace inequalities in several matrix or…

In 2008, M. Marshall settled a long-standing open problem by showing that if f(x,y) is a polynomial that is non-negative on the strip [0,1] x R, then there exist sums of squares s(x,y) and t(x,y) such that f(x,y) = s(x,y) + (x - x^2)…

We consider symmetric polynomials, p, in the noncommutative free variables (x_1, x_2, ..., x_g). We define the noncommutative complex hessian of p and we call a noncommutative symmetric polynomial noncommutative plurisubharmonic if it has a…

In this paper we shall show that there exists L_0 such that for each even integer L >= L_0 there exists $c_1 \in \rz$ for which the Julia set of $z --> z^L + c_1$ has positive Lebesgue measure. This solves an old problem. Editor's note: In…

This paper solves the rational noncommutative analog of Hilbert's 17th problem: if a noncommutative rational function is positive semidefinite on all tuples of hermitian matrices in its domain, then it is a sum of hermitian squares of…

There have been some effective tools for solving (constant/parametric) semi-algebraic systems in Maple's library RegularChains since Maple 13. By using the functions of the library, e.g., RealRootClassfication, one can prove and discover…

We consider the set of the power non-negative polynomials of several variables and its subset that consists of polynomials which can be represented as a sum of squares. It is shown in the classic work by D.Hilbert that it is a proper…

A linear map between real symmetric matrix spaces is positive if all positive semidefinite matrices are mapped to positive semidefinite ones. A real symmetric matrix is separable if it can be written as a summation of Kronecker products of…

We introduce the concept of disjunctive sum of squares for certifying nonnegativity of polynomials. Unlike the popular sum of squares approach where nonnegativity is certified by a single algebraic identity, the disjunctive sum of squares…

This paper concerns matrix "convex" functions of (free) noncommuting variables, $x = (x_1, \ldots, x_g)$. Helton and McCullough showed that a polynomial in $x$ which is matrix convex is of degree two or less. We prove a more general result:…

Let $F$ be an algebraically closed field of characteristic zero. We consider the question which subsets of $M_n(F)$ can be images of noncommutative polynomials. We prove that a noncommutative polynomial $f$ has only finitely many similarity…

With every real polynomial $f$, we associate a family $\{f_{\epsilon r}\}_{\epsilon, r}$ of real polynomials, in explicit form in terms of $f$ and the parameters $\epsilon>0,r\in N$, and such that $\Vert f-f_{\epsilon r}\Vert_1\to 0$ as…

Consider the closed convex hull $K$ of a monomial curve given parametrically as $(t^{m_1},\ldots,t^{m_n})$, with the parameter $t$ varying in an interval $I$. We show, using constructive arguments, that $K$ admits a lifted semidefinite…