Sums of squares in commutative ringsSpeaker: Professor Detlev Hoffmann (TU Dortmund)
Date: Thursday February 21st 2013
Location: Mathematics Seminar Room, Room AG 1.01, First Floor, Agriculture Building, UCD Belfield
We investigate several invariants related to sums of squares in commutative rings. The level is the smallest positive integer n such that -1 can be written as a sum of n squares. Questions regarding the level have been studied for about a century by Hilbert, van der Waerden, Artin, Schreier, Kaplansky, Pfister, Dai and Lam, to name but a few. Classically, the sublevel is defined to be the smallest positive integer n such that 0 can be written as a sum of n+1 squares in a unimodular way, i.e. that the elements that are to be squared generate as ideal the whole ring. This unimodularity condition is trivially satisfied in fields but it makes the study in arbitrary commutative rings difficult. A closer look reveals that it is useful to refine the notion of sublevel and to introduce what we call hyperlevel and metalevel. We study the relationship between these different notions of level and give some examples and discuss some open problems. This is joint work with David Leep.