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Every symmetric polynomial p(x)=p(x_1,...,x_g) (with real coefficients) in gnoncommuting variables x_1, ..., x_g can be written as a sum and difference ofsquares of noncommutative polynomials. Let s(p), the negative signature of p,denote the minimum number of negative squares used in this representation, andlet 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 noncommutativepolynomials p(x) such that s(p'') is 0 or 1 . We also introduce the relaxedHessian 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 andshow that if this relaxed Hessian is positive semidefinite in a suitable andrelatively innocuous way, then p has degree at most 2. Here R(x)[h] is a simpleuniversal positive polynomial which is quadratic in h. This analysis ismotivated by an attempt to develop properties of noncommutative real algebraicvarieties pertaining to their curvature, since, as will be shown elsewhere, - <p''_{L,K}(x)[h]v, v > (appropriately restricted) plays the role of of anoncommutative second fundamental form.

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