The Hessian of a noncommutative polynomial has numerous negative eigenvalues

In this paper, we establish bounds on the degree of a symmetric polynomial p = p(x) = p(x ₁,..., x _g ) (with real coefficients) in g noncommuting (nc) variables x ₁,..., x _g in terms of the “signature” of its Hessian which is a polynomial in x and h = (h ₁,..., h _g ) homogeneous of degree 2 in h. The bounds are obtained by exploiting the interplay between assorted representations for p(x) and p″(x)h] that are developed in the paper. In particular, p″(x)h] admits a representation of the form where f _j ⁺ , f _j ⁻ are nc polynomials. Such representations are highly non-unique. However, there is a unique smallest number of positive (resp., negative) squares σ _± ^min required in an SDS decomposition of p″(x)h]. Our main results yield the following corollary and a number of refinements. Supported by a Jay and Renee Weiss Chair. Partly supported by the NSF and the Ford Motor Co. Partly supported by the NSF grants DMS-0140112 and DMS-0457504.