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: (SDS) p(x) = Sigma sigma+/j=1 f(j)(+)(x)(T)f(j)(+)(x) - Sigma sigma-/l=1f(e)(-)(x)(T)f(l)(-)(x), where f(j)(+), f(l)(--) are noncommutative polynomials. Let sigma(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 p ''(x)[h] := d(2)p(x+th)/dt(2)vertical bar t=0. In this paper, we classify all symmetric noncommutative polynomials p(x) such that sigma(min)(-) (p '') <= 1. We also introduce the relaxed Hessian of a symmetric polynomial p of degree d via the formula p(lambda)'',(delta)(x)[h] := p ''(x)[h]+delta Sigma m(x)(T)h(j)(2)m(x)+lambda p '(x)[h](T)p '(x)[h] for lambda, delta, is an element of, 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, -< p(lambda)'',(delta)(x)[h]v,v > (appropriately restricted) plays the role of a noncommutative second fundamental form