2,438 research outputs found
A hermitian analogue of the Broecker-Prestel theorem
The Broecker-Prestel local-global principle characterizes weak isotropy of
quadratic forms over a formally real field in terms of weak isotropy over the
henselizations and isotropy over the real closures of that field. A hermitian
analogue of this principle is presented for algebras of index at most two. An
improved result is also presented for algebras with a decomposable involution,
algebras of pythagorean index at most two, and algebras over SAP and ED fields.Comment: Final pre-publication versio
Quadratic Forms and Space-Time Block Codes from Generalized Quaternion and Biquaternion Algebras
In the context of space-time block codes (STBCs), the theory of generalized
quaternion and biquaternion algebras (i.e., tensor products of two quaternion
algebras) over arbitrary base fields is presented, as well as quadratic form
theoretic criteria to check if such algebras are division algebras. For base
fields relevant to STBCs, these criteria are exploited, via Springer's theorem,
to construct several explicit infinite families of (bi-)quaternion division
algebras. These are used to obtain new 2\x 2 and 4\x 4 STBCs.Comment: 8 pages, final versio
The Procesi-Schacher conjecture and Hilbert's 17th problem for algebras with involution
In 1976 Procesi and Schacher developed an Artin-Schreier type theory for
central simple algebras with involution and conjectured that in such an algebra
a totally positive element is always a sum of hermitian squares. In this paper
elementary counterexamples to this conjecture are constructed and cases are
studied where the conjecture does hold. Also, a Positivstellensatz is
established for noncommutative polynomials, positive semidefinite on all tuples
of matrices of a fixed size.Comment: Final pre-publication versio
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