Sums of squares invariants for commutative rings

Expositor: Detlev Hoffmann Technische Universitaet Dortmund, Alemania

Jueves 21 de agosto a las 16:00 hrs.
Sala de magíster.

Resumen: The study of sums of squares in commutative rings has a long history. Typical questions are, for example, which elements can be written as a sum of squares, and how many squares are then actually needed. This naturally leads to certain invariants that one can attach to a commutative ring R. If an element x in R can be written as a sum of n squares but not fewer, we say it has length n (or infinity if x is not a sum of squares). The length of -1 is called the level of R, and the Pythagoras number of R is the supremum of the lengths of all elements that are sums of squares. For example, a well known result due to Lagrange states that the ring of integers has Pythagoras number equal to 4. There is also the notion of sublevel, the least n such that 0 can be written as a sum of n+1 squares of elements that generate the ring as an ideal (or infinity if no such n exists). Level, sublevel and Pythagoras number are intimately related. We give a survey of some old and some new results and highlight some of the many open problems.