We give a sufficient condition for a model theoretic structure B to ‘inherit’ quantifier elimination from another structure A . This yields an alternative proof of one of the main results from [8] , namely quantifier elimination for certain matrix rings. The original proof uses model theory, and while it is very elegant and insightful, the proof we propose is much shorter and provides a constructive algorithm.

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 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 ε -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.