| Résume||Joint work with F. Miraglia and H. Ribeiro (University of São Paulo, Brazil).
This talk will present (in English or French, as the public wishes) a summary of the main results of a joint paper (« Special groups and quadratic forms over rings with non zero-divisor coefficients », 60 pp.), to appear in Fundamenta Math. We develop a theory of quadratic forms with non zero-divisor coefficients over preordered (commutative, unitary) rings , where 2 is invertible and the preorder T satisfies a mild extra condition. In this context we prove that several major results known to hold in classical quadratic form theory over (orderable) fields –e.g., the Arason-Pfister Hauptsatz and Pfister’s localglobal principle-- carry over to any class of preordered rings satisfying a property called NT-quadratic faithfulness. We show that this property holds for many classes of rings frequently met in mathematical practice, such as:
— the reduced f-rings and some of their extensions, for which Marshall’s signature conjecture and a vast generalization of Sylvester’s inertia law hold as well;
— the reduced, partially ordered Noetherian rings and many of their quotients (a result relevant in real algebraic geometry).
The abstract, axiomatic approach to quadratic form theory known as special groups [Dickmann-Miraglia, Memoirs AMS 689 (2000)] is the main tool used in the proofs of these results.|