Approximate Pythagoras Numbers on $*$-algebras over $\mathbb{C}$

TL;DR

This paper investigates how the approximate Pythagoras number in $*$-algebras over $\

Contribution

It introduces the concept of approximate Pythagoras numbers and demonstrates they can be significantly smaller and more manageable than exact numbers, with bounds independent of dimension.

Findings

Approximate Pythagoras numbers are often much smaller than exact ones.

Small perturbations can significantly reduce the Pythagoras number.

The study provides dimension-independent upper bounds for approximate numbers.

Abstract

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.