Hessian具有负符号1和非交换第二基本型的非交换多项式的分类
Classification of All Noncommutative Polynomials Whose Hessian Has Negative Signature One and A Noncommutative Second Fundamental Form
摘要：
每个对称多项式p（x）=p（x_1，.，x_g）（具有实系数）在g个非交换变量x_1，...，x_g可以写成非交换多项式的平方和与平方差。设s（p），p的负签名，表示在该表示中使用的负平方的最小数目，并且设p的非交换海森由公式p“（x）[h]定义：= d^2p（x+th）\dt^2|_{t=0}。本文对所有s（p“）为0或1的对称非交换多项式p（x）进行了分类.我们还通过公式p“_{L，K}（x）[h]：= p”（x）[h] + Lp“（x）[h]^{T} p "（x）[h] + KR（x）[h]对L，K实数引入了d次对称多项式p的松弛Hessian，并证明了如果该松弛Hessian以适当且相对无害的方式是半正定的，则p的次数至多为2.这里R（x）[h]是一个简单的泛正多项式，它在h中是二次的。这种分析的动机是试图发展与其曲率有关的非交换实代数簇的性质，因为正如在其他地方所示，- <p“_{L，K}（x）[h]v，v >（适当限制）扮演着非交换第二基本型的角色。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
