The Hessian of a noncommutative polynomial has numerous negative eigenvalues

Abstract

In this paper, we establish bounds on the degree of a symmetric polynomial p = p(x) = p(x1,..., xg) (with real coefficients) in g noncommuting (nc) variables x1,..., xg in terms of the “signature” of its Hessian $$p''(x)[h]: = \frac{{d^2 p(x + th)}}{{dt^2 }}|t = 0,$$ which is a polynomial in x and h = (h1,..., hg) 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 \((SDS) p''(x)[h] = \sum\limits_{j = 1}^{\sigma _ + } {f_j^ + (x)[h]^T f_j^ + (x)[h]} - \sum\limits_{\ell = 1}^{\sigma _ - } {f_\ell ^ - (x)[h]^T f_\ell ^ - (x)[h]} \) where fj+, fj− 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.