A Positivstellensatz for Non-commutative Polynomials

Positivstellensatz and Flat Functionals on Path ∗-algebras

We consider the class of non-commutative ∗-algebras which are path algebras of doubles of quivers with the natural involutions. We study the problem of extending positive truncated functionals on such ∗-algebras. An analog of the solution of the truncated Hamburger moment problem [Fia91] for path ∗-algebras is presented and non-commutative positivstellensatz is proved. We aslo present an analog...

A Non-Commutative Positivstellensatz On Isometries

Real Spectrum and the Abstract Positivstellensatz

Algebraic geometry emerged from the study of subsets of C defined by polynomial equations, namely algebraic sets. Parallel to this, real algebraic geometry emerged from studying subsets of R defined by polynomial equations and inequalities, namely semi-algebraic sets. Like algebraic geometry, classical methods in real algebraic geometry provided useful but limited intuition on the relationship ...

An algorithmic approach to Schmüdgen’s Positivstellensatz

We present a new proof of Schmüdgen’s Positivstellensatz concerning the representation of polynomials f ∈ R[X1, ..., Xd] that are strictly positive on a compact basic closed semialgebraic subset S of Rd. Like the two other existing proofs due to Schmüdgen and Wörmann, our proof also applies the classical Positivstellensatz to non–constructively produce an algebraic evidence for the compactness ...

On an extension of Pólya's Positivstellensatz

In this paper we provide a generalization of a Positivstellensatz by Pólya [Georg Pólya. Über positive darstellung von polynomen vierteljschr. In Naturforsch. Ges. Zürich, 73: 141-145, 1928]. We show that if a homogeneous polynomial is positive over the intersection of the non-negative orthant and a given basic semialgebraic cone (excluding the origin), then there exists a “Pólya type” certific...