A univariate trace polynomial is a polynomial in a variable x and formal trace symbols $\mathrm{Tr}(x^j)$. Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as normalized traces. This paper addresses global and constrained positivity of univariate trace polynomials on symmetric matrices of all finite sizes. A tracial analog of Artin's solution to Hilbert's 17th problem is given: a positive semidefinite univariate trace polynomial is a quotient of sums of products of squares and traces of squares of trace polynomials.

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正单变量跟踪多项式

单变量跟踪多项式是变量x和形式跟踪符号中的多项式$\mathrm{Tr}（X^{Ĵ}）$。这样的表达式可以自然地在矩阵上求值，其中的迹线符号被评估为归一化迹线。本文讨论了所有有限大小的对称矩阵上单变量跟踪多项式的全局正负约束。给出了Artin解决希尔伯特（Hilbert）第17个问题的方法的类比：正半定单变量跟踪多项式是平方乘积和跟踪多项式平方和的和的商。