Propriétés géométrico-arithmétiques des algèbres à involution. (Geometrico-arithmetic properties of algebras with involution) (Q1101498)

The author develops a theory of quadratic forms and algebras with involution over certain Euclidean rings. An old result appearing in {\S} 6.8 of [\textit{J. W. S. Cassels}, Rational quadratic forms (Academic Press 1978; Zbl 0395.10029)], states that if f(x) is an isotropic quadratic form in n variables over the ring of integers, there is an isotropic vector \((a_ 1,a_ 2,...,a_ n)\) where \(\max \{| a_ j| \}\) is bounded. In his Théorème 6 the author generalizes this bound to the case of forms over k[t], where k is a field of characteristic not 2. There is some overlap here with \textit{A. Prestel} [J. Reine Angew. Math. 378, 101-112 (1987; Zbl 0606.10014)]. A generalized version of the Cassels-Pfister theorem for quadratic forms over \(k[t]\) is proved in Théorème 7. Note that the ring there is still k[t]. There is a minor error in Lemma 2, which proliferates in several subsequent statements. Namely if A is an algebra with involution over a field R and an element x has \(x\bar x=0\), it does not follow that \(\bar xx=0\). This error requires some restatements of Propositions 2 and 4, but it does not affect the proofs of the theorems.