Classification of all noncommutative polynomials whose Hessian has negative signature one and a noncommutative second fundamental form

Every symmetric polynomial p = p ( x ) = p ( x 1 ,..., x g ) (with real coefficients) in g noncommuting variables x 1 ,..., x g can be written as a sum and difference of squares of noncommutative polynomials: where f j + , f ℓ − are noncommutative polynomials. Let σ − min ( p ), the negative signatu...

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Veröffentlicht in:Journal d'analyse mathématique (Jerusalem) 2009-05, Vol.108 (1), p.19-59
Hauptverfasser: Dym, Harry, Greene, Jeremy M., Helton, J. William, McCullough, Scott A.
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Sprache:eng
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Zusammenfassung:Every symmetric polynomial p = p ( x ) = p ( x 1 ,..., x g ) (with real coefficients) in g noncommuting variables x 1 ,..., x g can be written as a sum and difference of squares of noncommutative polynomials: where f j + , f ℓ − are noncommutative polynomials. Let σ − min ( p ), the negative signature of p , denote the minimum number of negative squares used in this representation; and let the Hessian of p be defined by the formula In this paper, we classify all symmetric noncommutative polynomials p ( x ) such that We also introduce the relaxed Hessian of a symmetric polynomial p of degree d via the formula for λ, δ, ∈, ℝ and show that if this relaxed Hessian is positive semidefinite in a suitable and relatively innocuous way, then p has degree at most 2. Here the sum is over monomials m ( x ) in x of degree at most d − 1 and 1 ≤ j ≤ g . This analysis is motivated by an attempt to develop properties of noncommutative real algebraic varieties pertaining to curvature, since, as will be shown elsewhere, (appropriately restricted) plays the role of a noncommutative second fundamental form.
ISSN:0021-7670
1565-8538
DOI:10.1007/s11854-009-0017-0