正迹多项式与普适Procesi-Schacher猜想
Positive trace polynomials and the universal Procesi-Schacher conjecture
摘要：
正性定理是实代数几何中的一个基本结果，它为半代数集上的多项式的正性提供了代数证明。在这篇文章Positivstellensätze为迹多项式正半代数集的$n\times n$矩阵提供。本文利用厄米特平方和算子的加权和，证明了一个Krivine-Stengle型正定矩阵，它刻画了一般半代数集$K$上迹多项式的非负性.这些证书中的权重是从$K$的生成元和厄米平方的迹线中获得的。对于紧半代数集$K$，得到了Schmüdgen-和Putinar-型Positivstellensätze：在$K$上的迹多项式正有一个带权和不带分解子的Hermitian平方和分解.所采用的方法的灵感来自不变量理论，经典的实代数几何和泛函分析。Procesi和Schacher在1976年发展了一个关于对合中心单代数的序和正性的理论，并对$n$度的泛中心单代数提出了一个Hilbert第17问题：每个全正元素是埃尔米特平方和吗？他们对$n=2$给出了肯定的回答。本文对$n=3$给出了否定的回答。因此，在Positivstellensätze中包含埃尔米特平方的迹作为权重是必不可少的。
英文摘要：
Positivstellensatz is a fundamental result in real algebraic geometry providing algebraic certificates for positivity of polynomials on semialgebraic sets. In this article Positivstellensätze for trace polynomials positive on semialgebraic sets of $n\times n$ matrices are provided. A Krivine-Stengle-type Positivstellensatz is proved characterizing trace polynomials nonnegative on a general semialgebraic set $K$ using weighted sums of hermitian squares with denominators. The weights in these certificates are obtained from generators of $K$ and traces of hermitian squares. For compact semialgebraic sets $K$ Schmüdgen- and Putinar-type Positivstellensätze are obtained: every trace polynomial positive on $K$ has a sum of hermitian squares decomposition with weights and without denominators. The methods employed are inspired by invariant theory, classical real algebraic geometry and functional analysis. Procesi and Schacher in 1976 developed a theory of orderings and positivity on central simple algebras with involution and posed a Hilbert's 17th problem for a universal central simple algebra of degree $n$: is every totally positive element a sum of hermitian squares? They gave an affirmative answer for $n=2$. In this paper a negative answer for $n=3$ is presented. Consequently, including traces of hermitian squares as weights in the Positivstellensätze is indispensable.
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。
