$\mathbb{C}$上$*$ -代数上的近似Pythagoras数
Approximate Pythagoras Numbers on $*$-algebras over $\mathbb{C}$
摘要：
平方和的毕达哥拉斯数是平方和表示中长度最短的。在许多代数中，例如两个或多个变量的实多项式代数，对于所有平方和，不存在毕达哥拉斯数的上界。本文研究了$\mathbb{C}$上$*$ -代数中的毕达哥拉斯数对元素的小扰动的行为。更准确地说，一个元素的近似毕达哥拉斯数是它的$\varepsilon$ -球中所有元素中最小的毕达哥拉斯数。我们发现，这些近似的毕达哥拉斯数往往是显着小于他们的确切版本，并允许（几乎）尺寸无关的上界。我们的结果使用低秩近似的革兰氏矩阵的平方和和估计的操作规范的革兰氏映射。
英文摘要：
The Pythagoras number of a sum of squares is the shortest length among its sums of squares representations. In many algebras, for example real polynomial algebras in two or more variables, there exists no upper bound on the Pythagoras number for all sums of squares. In this paper, we study how Pythagoras numbers in $*$-algebras over $\mathbb{C}$ behave with respect to small perturbations of elements. More precisely, the approximate Pythagoras number of an element is the smallest Pythagoras number among all elements in its $\varepsilon$-ball. We show that these approximate Pythagoras numbers are often significantly smaller than their exact versions, and allow for (almost) dimension-independent upper bounds. Our results use low-rank approximations for Gram matrices of sums of squares and estimates for the operator norm of the Gram map.
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