  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.
