M/E23

For any ring R, its Pythagoras number is the smallest number P(R) such that any sum of squares in R can be written as a sum of at most P(R) squares. Lagrange's celebrated four-square theorem can be stated as P(Z)=4. We will look for analogous results in other number fields; the most interesting ones turn out to be totally real fields, i.e. such that the image of all their complex embeddings is in fact a subset of the real numbers. While the Pythagoras number of a number field is easy to determine, the Pythagoras number of its ring of integers is usually unknown. We will discuss the available results and the basic ideas behind them. In particular, to obtain upper bounds for Pythagoras numbers, we introduce the so-called g-invariants, which are similar to the Pythagoras number, but squares of numbers are replaced by squares of linear forms. The study of g-invariants, sometimes called the quadratic Waring's problem, is far from solved even in the case of rational integers; however, we will see some nontrivial results.

Jakub Krásenský

Prag