On quadratic forms whose total signature is zero mod \(2^n\). Solution to a problem of M. Marshall.

*(English)*Zbl 0908.11022The main aim of the paper is to prove that if \(F\) is a formally real pythagorean field and the signature sgn\(_P(\phi)\) of the quadratic form \(\phi\) over \(F\) is zero modulo \(2^n\) for every order \(P\) of \(F\), then \(\phi\in I^n(F)\), where \(I(F)\) is the fundamental ideal of the Witt ring \(W(F)\). It solves the problem posed by M. Marshall [|Proc. conf. on quadratic forms, Queen’s Pap. Pure Appl. Math. 46, 569-579 (1977; Zbl 0385.12008)].

The paper is divided into two parts. In the first part the authors reformulate the original problem into the so-called Weak Marshall Conjecture, which is given an affirmative answer in the second part. The main result depends on Voevodsky’s recent result on Milnor homomorphisms. In the first part the authors work in the framework of so called special groups and related Boolean-theoretic techniques. The theory of special groups is introduced in their forthcoming paper and resembles a modified version of Marshall’s abstract theory of quadratic forms. For the convenience of the reader, they recall the basic notions and facts of this theory.

The paper is divided into two parts. In the first part the authors reformulate the original problem into the so-called Weak Marshall Conjecture, which is given an affirmative answer in the second part. The main result depends on Voevodsky’s recent result on Milnor homomorphisms. In the first part the authors work in the framework of so called special groups and related Boolean-theoretic techniques. The theory of special groups is introduced in their forthcoming paper and resembles a modified version of Marshall’s abstract theory of quadratic forms. For the convenience of the reader, they recall the basic notions and facts of this theory.

Reviewer: A.Sładek (Katowice)

##### MSC:

11E81 | Algebraic theory of quadratic forms; Witt groups and rings |

11E04 | Quadratic forms over general fields |

12D15 | Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) |