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

TL;DR

This paper classifies symmetric noncommutative polynomials with Hessians having at most one negative eigenvalue, and introduces a relaxed Hessian to analyze polynomial degree and curvature properties in noncommutative algebraic geometry.

Contribution

It provides a complete classification of polynomials with Hessian negative signature one and introduces a relaxed Hessian to study polynomial degree and curvature.

Findings

Classified all symmetric noncommutative polynomials with Hessian negative signature 0 or 1.

Showed that positive semidefinite relaxed Hessians imply polynomial degree at most 2.

Developed a framework connecting Hessian signatures to noncommutative curvature properties.

Abstract

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.