The second edition of the summer school ALGAR is dedicated to the study of sums of squares in fields. In four lecture series and a few special talks we introduce the audience to the necessary algebraic, arithmetic and geometric methods that yield to beautiful and deep results on the length of sums of squares in certain types of fields.
Sums of squares have been studied since antiquity. The development of classical algebraic number theory was accompanied by famous achievements by Fermat, Euler, Lagrange and Gauss on the representation of natural numbers as sums of squares.
Writing polynomials with real coefficients as sums of squares is a way of showing that they are positive definite. This is also relevant in applied mathematics, for example in linear optimisation. In his 17th problem Hilbert asked whether any positive definite rational function with real coefficients in n variables can be written as a sum of squares of rational functions. He had shown this for n=1 and n=2. The general positive answer was obtained by Artin in 1927, but its proof is not constructive and does not yield directly an upper bound on the number of squares that is required. However, in 1967 Pfister showed that a representation as a sum of 2^n squares of rational functions with real coefficients is always possible.
The Pythagoras number of a commutative ring is defined as the smallest number p such that every sum of squares in the ring is equal to a sum of p squares. It can be very difficult to determine the Pythagoras number of a given ring. In particular, it is very hard to show that a certain sum of squares cannot be written with fewer terms. In the summer school, we will restrict mostly to fields, with a particular focus on rational function fields, function fields of real surfaces and function fields over the rational numbers.
The aim of the event is to survey some of the methods that are used in the study of sums of squares and thereby to illustrate various relations of this active research area to other branches of mathematics, obviously including algebraic geometry and number theory, but also analysis and topology, and furthermore logic and model theory.