74 citations · 668 across the 48 of their papers we have counts for
5 papers · 1 filter
Matrix Convex Hulls of Free Semialgebraic Sets
J. William Helton, Igor Klep, Scott McCullough
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…
Noncommutative polynomials nonnegative on a variety intersect a convex set
J. William Helton, Igor Klep, Christopher S. Nelson
By a result of Helton and McCullough, open bounded convex free semialgebraic sets are exactly open (matricial) solution sets D_L of a linear matrix inequality (LMI) L(X)>0. This pa…
Free Semidefinite Representation of Matrix Power Functions
J. William Helton, Jiawang Nie, Jeremy S. Semko
Consider the matrix power function X^p defined over the cone of positive definite matrices S^{n}_{++}. It is known that X^p is convex over S^{n}_{++} if p is in [-1,0] or [1,2] and…
Free Convex Algebraic Geometry
J. William Helton, Igor Klep, Scott McCullough
This chapter is a tutorial on techniques and results in free convex algebraic geometry and free real algebraic geometry (RAG). The term free refers to the central role played by al…
Real Nullstellensatze and *-ideals in *-algebras
Jakob Cimpric, J. William Helton, Scott McCullough +1
Let F denote either the real or complex field. An ideal I in the free *-algebra F<x,x*> in g freely noncommuting variables and their formal adjoints is a *-ideal if I = I*. When a…