IUMJ

Title: Irreducible noncommutative defining polynomials for convex sets have degree 4 or less

Authors: Harry Dym, J. William Helton and Scott McCullough

Issue: Volume 56 (2007), Issue 3, 1189-1232

Abstract:

A non-commutative polynomial $p(x_1, \dots , x_g)$ is a linear combination of words in the non-commuting variables $\{x_1, \dots ,x_g\}$. Such a polynomial is naturally evaluated on a tuple $X = (X_1, \dots ,X_g)$ of symmetric $n \times n$ matrices, with value $p(X)$ an $n \times n$ matrix. The involution ${}^T$ on words given by sending a concatenation of letters to the same letters, but in the reverse order (for instance $(x_j x_{\ell})^T = x_{\ell} x_j)$ extends naturally to such polynomials and $p$ is itself symmetric if $p^T = p$. In this case, $p(X)$ is a symmetric matrix. The positivity domain $\mathcal{D}_{p}^n$ of a non-commutative symmetric polynomial $p$ is the closure of the component of $0$ of the set \[ \{X \in (\mathbb{R}^{n \times n}_{\mathrm{sym}})^{g} \mid p(X) \succ 0\}. \] Here $(\mathbb{R}^{n \times n}_{\mathrm{sym}})^{g}$ denotes the set of $g$-tuples of $n \times n$ real symmetric matrices. The positivity domain, $\mathcal{D}_{p}$, is the sequence of sets $\{\mathcal{D}_{p}^n\}$.\par The purpose of this paper is to prove that, under some additional hypotheses on $p$, the convexity of the set $\mathcal{D}_p$ plus the irreducibility (in an appropriate sense) of $p$ imply that degree of $p$ is at most four and that $p$ has additional structure, which is also discussed in detail.\par This result may portend a type of noncommutative (in a free algebra) real algebraic geometry in which basic conditions on a variety $V$ constrain $V$ much more than occurs classically. Here an irreducible noncommutative variety (namely the boundary of $\mathcal{D}_p$) with nonnegative curvature has degree no greater than four. The problem itself is motivated by linear system engineering and the vast quantity of work there on Linear Matrix Inequalities (LMIs) and Convex Matrix Inequalities. It suggests that in systems problems, whose form scales with dimension, convex situations are very heavily constrained. This paper treats the geometry of noncommutative varieties, whereas earlier work (J. William Helton and Scott A. McCullough, \emph{Convex noncommutative polynomials have degree two or less}, SIAM J. Matrix Anal. Appl. \textbf{25} (2004), 1124--1139; J. William Helton, Scott A. McCullough, and Victor Vinnikov, \emph{{Noncommutative convexity arises from linear matrix inequalities}, J. Funct. Anal. \textbf{240} (2006), 105--191) treats "convex" noncommutative polynomials and rational functions.\par Our approach here includes an analysis of non-commutative second directional derivatives $p''(x)[h]$, a non-commutative polynomial in $2g$ variables, with respect to the number of positive and negative eigenvalues of $p''(X)[H]$ for $X$, $H \in (\mathbb{R}^{n \times n}_{\mathrm{sym}})^{g}$. The analysis in the present paper is for $X$ in the boundary of $\mathcal{D}_{p}$, and $H$ corresponding to directions tangent to the boundary of $\mathcal{D}_p$, restrictions which cause very many difficulties. The case where $X$ is not constrained is treated in Harry Dym, J. William Helton, and Scott A. McCullough, \emph{The Hessian of a noncommutative polynomial has numerous negative eigenvalues}, Journal d'Analyse Mathematique (to appear).