Sums of squares in certain quaternion and octonion algebras

Formulae for the levels and sublevels of certain quaternion and octonion algebras are established. Corollaries concerning the equality of levels and sublevels of quaternion algebras with those of associated octonion algebras are presented.     

Sums of squares in nonreal commutative rings

We consider the problem of finding a zero of an accretive operator in a Banach space and prove strong convergence results for resolvents of the accretive operator.     

Sums of two squares in analytic rings

We study analytic singularities for which every positive semidefinite analytic function is a sum of two squares of analytic functions. This is a basic useful property of the plane, but difficult to check in other cases; in particular, what about , , or ? In fact, the unique positive examples we can find are the Brieskorn singularity, the union of two planes in 3-space and the Whitney umbrella. Conversely, we prove that a complete intersection with that property (other than the seven embedded surfaces already mentioned) must be a very simple deformation of the two latter, namely, In particular, except for the stems and , all singularities are real rational double points. Received April 4, 1997; in final form September 25, 1997     

Sums of corestrictions of cyclic algebras

By a cyclic layer of a finite Galois extension,E/K, of fields one means a cyclic extension,L/F, of fields whereE⊇L⊇F⊇K. LetC(E/K) denote the subgroup of the relative Brauer group, Br(E/K), generated by the various subgroups cor(Br(L/F)) asL/F ranges over all cyclic layers ofE/K and where cor denotes the corestriction map into Br(E/K). We show that forK global, [Br(E/K) :C(E/K)]<∞ and we produce examples whereC(E/K)≠Br(E/K). In memory of S.A. Amitsur, our teacher, friend, collaborator, and inspiration. Supported in part by NSA Grant No. MDA904-95-H-1054. Supported in part by NSA Grant No. MDA904-95-H-1022.     

Sums of squares on real algebraic surfaces

Consider real polynomials g₁, . . . , g_r in n variables, and assume that the subset K = {g₁≥0, . . . , g_r≥0} of ℝⁿ is compact. We show that a polynomial f has a representation in which the s_e are sums of squares, if and only if the same is true in every localization of the polynomial ring by a maximal ideal. We apply this result to provide large and concrete families of cases in which dim (K) = 2 and every polynomial f with f|_K≥0 has a representation (*). Before, it was not known whether a single such example exists. Further geometric and arithmetic applications are given. Support by DFG travel grant KON 1823/2002 and by the European RAAG network HPRN-CT-2001-00271 is gratefully acknowledged. Part of this work was done while the author enjoyed a stay at MSRI Berkeley. He would like to thank the institute for the invitation and the very pleasant working conditions.     

Sums of squares on real algebraic curves

Given an affine algebraic variety V over with real points V() compact and a non-negative polynomial function f[V] with finitely many real zeros, we establish a local-global criterion for f to be a sum of squares in [V]. We then specialize to the case where V is a curve. The notion of virtual compactness is introduced, and it is shown that in the local-global principle, compactness of V() can be relaxed to virtual compactness. The irreducible curves on which every non-negative polynomial is a sum of squares are classified. All results are extended to the more general framework of preorders. Moreover, applications to the K-moment problem from analysis are given. In particular, Schmüdgens solution of the K-moment problem for compact K is extended, for dim (K)=1, to the case when K is virtually compact. Mathematics Subject Classification (1991):14P05, 11E25, 14H99, 14P10, 44A60. Dedicated to Eberhard Becker on the occasion of his 60th birthday     

Sums of squares on reducible real curves

We ask whether every polynomial function that is non-negative on a real algebraic curve can be expressed as a sum of squares in the coordinate ring. Scheiderer has classified all irreducible curves for which this is the case. For reducible curves, we show how the answer depends on the configuration of the irreducible components and give complete necessary and sufficient conditions. We also prove partial results in the more general case of finitely generated preorderings and discuss applications to the moment problem for semialgebraic sets.     

Selectivity in division algebras

A commutative order in a central simple algebra over a number field is said to be selective if it embeds in some, but not all, maximal orders in the algebra. We completely characterize selective orders in central division algebras, of dimension 9 or greater, in terms of the characterization of selective orders given by Chinburg and Friedman in the quaternionic case.     

Sums of squares in function fields of hyperelliptic curves

We study sums of squares, quadratic forms, and related field invariants in a quadratic extension of the rational function field in one variable over a hereditarily pythagorean base field.     

Cyclic division algebras

A general example of cyclic division algebra is given, based on a construction of Brauer, yielding examples of division algebras of arbitrary prime exponent without proper central subalgebras, and also noncrossed products of arbitrary exponent. This research was supported in part by the U.S.-Israel Binational Science Foundation. An erratum to this article is available at .     

Graded division algebras

Mathematische Zeitschrift -     

Sums of squares in imaginary quadratic fields. II

This article presents new quantitative results associated with the representation of integers as the sums of three squares in imaginary quadratic fields. Errors made in Part I are corrected.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 160–167, 1990.