Illinois Journal of Mathematics

Non-commutative varieties with curvature having bounded signature

Harry Dym, J. William Helton, and Scott McCullough

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A natural notion for the signature $C_{\pm}({\mathcal V}(p))$ of the curvature of the zero set ${\mathcal V}(p)$ of a non-commutative polynomial $p$ is introduced. The main result of this paper is the bound \[ \operatorname{deg} p \leq2 C_\pm \bigl({\mathcal V}(p) \bigr) + 2. \] It is obtained under some irreducibility and nonsingularity conditions, and shows that the signature of the curvature of the zero set of $p$ dominates its degree.

The condition $C_+({\mathcal V}(p))=0$ means that the non-commutative variety ${\mathcal V}(p)$ has positive curvature. In this case, the preceding inequality implies that the degree of $p$ is at most two. Non-commutative varieties ${\mathcal V}(p)$ with positive curvature were introduced in Indiana Univ. Math. J. 56 (2007) 1189-1231). There a slightly weaker irreducibility hypothesis plus a number of additional hypotheses yielded a weaker result on $p$. The approach here is quite different; it is cleaner, and allows for the treatment of arbitrary signatures.

In J. Anal. Math. 108 (2009) 19-59), the degree of a non-commutative polynomial $p$ was bounded by twice the signature of its Hessian plus two. In this paper, we introduce a modified version of this non-commutative Hessian of $p$ which turns out to be very appropriate for analyzing the variety ${\mathcal V}(p)$.

Article information

Illinois J. Math., Volume 55, Number 2 (2011), 427-464.

First available in Project Euclid: 1 February 2013

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Zentralblatt MATH identifier

Primary: 47Axx: General theory of linear operators
Secondary: 47A63: Operator inequalities 47L07: Convex sets and cones of operators [See also 46A55] 47L30: Abstract operator algebras on Hilbert spaces 14P10: Semialgebraic sets and related spaces


Dym, Harry; Helton, J. William; McCullough, Scott. Non-commutative varieties with curvature having bounded signature. Illinois J. Math. 55 (2011), no. 2, 427--464. doi:10.1215/ijm/1359762396.

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