Signatures of hermitian forms and the Knebusch Trace Formula

Signatures of quadratic forms have been generalized to hermitian forms over algebras with involution. In the literature this is done via Morita theory, which causes sign ambiguities in certain cases. In this paper, a hermitian version of the Knebusch Trace Formula is established and used as a main tool to resolve these ambiguities. The last page is an erratum for the published version. We inadvertently (I) gave an incorrect definition of adjoint involutions; (II) omitted dealing with the case $(H\times H, \widehat{\phantom{m}}\,)$. As $W(H\times H, \widehat{\phantom{m}}\,)= W(R\times R, \widehat{\phantom{m}}\,)=0$, the omission does not affect our reasoning or our results. For the sake of completeness we point out where some small changes should be made in the published version.Comment: This is the final version before publication. The last page is an updated erratum for the published versio

Decompositions of modules lacking zero sums

The third author thanks the University of Virginia mathematics department for its hospitality.Peer reviewedPostprin

Quadratic forms of dimension 8 with trivial discrimiand and Clifford algebra of index 4

Izhboldin and Karpenko proved in 2000 that any quadratic form of dimension 8 with trivial discriminant and Clifford algebra of index 4 is isometric to the transfer, with respect to some quadratic \'etale extension, of a quadratic form similar to a 2-fold Pfister form. We give a new proof of this result, based on a theorem of decomposability for degree 8 and index 4 algebras with orthogonal involution

Motivic equivalence of affine quadrics

In this article we show that the motive of an affine quadric {q=1} determines the respective quadratic form

Linear spaces on the intersection of cubic hypersurfaces

Upper bounds for the number of variables necessary to imply the existence of an m -dimensional linear variety on the intersection of r cubic hypersurfaces over local and global fields are given.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41633/1/605_2006_Article_BF02349626.pd