Quadratic forms and Invariants
Final Report Abstract
This project investigates quadratic forms, algebras with involution, and related algebraic structures defined over fields, and it studies their properties in terms of invariants. This concerns invariants of the particular structure as well as invariants of the underlying field. The invariants can be numbers, algebraic objects, or just properties. The aim is to relate different invariants and to study their behavior under scalar extension. A major part of this project concerns field invariants taking values in N U {∞}. The most studied ones in this context are the pythagoras number, defined as the upper bound of the lengths of sums of squares in the field, and the u-invariant, defined as the maximal dimension of an anisotropic quadratic torsion form over the field. The focus is on the study of real fields. A good example of a relation between invariants is the estimate l ≤ 1/2 pu + 1 , obtained in collaboration with D. Leep, which bounds the length of a field in terms of the pythagoras number and the u-invariant. Several new estimates involving the length were found and further shown to be best possible by using generic splitting techniques and a field construction due to Merkurjev. A systematic study of the use of valuations in obtaining lower bounds on field invariants was carried out. Furthermore, known bounds on the length and the u-invariant of a function field over a real closed field were improved. Another field invariant that is interesting in the study of real fields and their spaces of orderings is the stability index. In a collaboration with P. Gladki, for a pythagorean field, where every sum of squares is a square, the finiteness of the stability index is shown to be equivalent to the finiteness of the 2-symbol length. In a joint work with D. Leep and C. Schubert the behavior of the stability index under various types of field extensions is investigated. A study of sums of squares in algebraic function fields was carried out as a PhD-project by D. Grimm. One result obtained is that, if in a function field of a conic every sum of squares is equal to a sum of two squares, then the base field either contains √-1 or it is real and all its finite real extensions are pythagorean. This result is derived as a consequence of a general result on residue fields of points on a geometrically rational variety. In a joint work with D. Grimm and J. Van Geel, a recently found local-global principle based on Field Patching is applied to prove results on sums of squares in algebraic function fields over a power series field. Two subprojects explore topics in Milnor K-theory. As a generalization of quadratic forms, virtual forms are introduced as a tool to study Milnor's K-theory via operations generalizing Stiefel-Whitney classes. In a collaboration with M. Raczek, the second Milnor K-group of a rational function field is studied by means of an exact sequence originally, given in Galois cohomology by Faddeev. The number of symbols needed to produce an element with a given ramification is bounded in terms of the ramification degree. Given the unavailability of a postdoctoral researcher specialized in algebra and model theory, the originally planned partial focus on Abstract Quadratic Form Theory was deemphasized in favor of including a study of algebras with involution in characteristic two. This research, carried out by A. Dolphin, provides new results on the metabolicity behavior of involutions under quadratic field extensions and a particular type of decomposition of hermitian forms and algebras with involution in characteristic two. In a joint work with T. Unger, algebras with involution over (formally) real fields are studied. Via a classification over real closed fields, a uniform treatment of signatures is given and a generalized version of Pfister's Local-Global Principle for algebras with involution is obtained.
Publications
- Pythagoras numbers and quadratic field extensions. Proceedings of the International Conference on the Algebraic and Arithmetic Theory of Quadratic Forms 2007, Lago Llanquihue (Chile). Contemporary Mathematics 493 (2009): 21-28
K. J. Becher and D. B. Leep
- On the u-invariant of a real function field. Mathematische Annalen 346 (2010): 245-249
K. J. Becher
- Virtual forms. Mathematische Zeitschrift 265 (2010): 551-569
K. J. Becher
- Metabolic involutions. Journal of Algebra 336 (2011): 286-300
A. Dolphin
- The length and other invariants of a real field. Mathematische Zeitschrift 269 (2011): 235-252
K. J. Becher and D. B. Leep
- Symbol length and stability index. Journal of Algebra, 354 (2012): 71-76
K. J. Becher and P. Gladki
