Classification of All Noncommutative Polynomials Whose Hessian Has Negative Signature One and A Noncommutative Second Fundamental Form

Details Classification of All Noncommutative Polynomials Whose Hessian Has Negative Signature One and A Noncommutative Second Fundamental Form Classification of All Noncommutative Polynomials Whose Hessian Has Negative Signature One and A Noncommutative Second Fundamental Form

2009-03-11

arxiv.org/abs/0903.2029v1

Mathematics

Functional Analysis

44 pages

Scientific paper

Every symmetric polynomial 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. Let s(p), the negative signature of p, denote the minimum number of negative squares used in this representation, and let the noncommutative Hessian of p be defined by the formula p''(x)[h] := d^2p(x+th)\dt^2|_{t=0}. In this paper we classify all symmetric noncommutative polynomials p(x) such that s(p'') is 0 or 1 . We also introduce the relaxed Hessian of a symmetric polynomial p of degree d via the formula p''_{L,K}(x)[h] := p''(x)[h] + L p'(x)[h]^{T} p'(x)[h] + K R(x)[h] for L, K real numbers 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 R(x)[h] is a simple universal positive polynomial which is quadratic in h. This analysis is motivated by an attempt to develop properties of noncommutative real algebraic varieties pertaining to their curvature, since, as will be shown elsewhere, - < p''_{L,K}(x)[h]v, v > (appropriately restricted) plays the role of of a noncommutative second fundamental form.

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