Sums of squares in real analytic rings

Let be an analytic ring. We show: (1) has finite Pythagoras number if and only if its real dimension is , and (2) if every positive semidefinite element of is a sum of squares, then is real and has real dimension .

     

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 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.     

Sums of distinct integral squares in real quadratic fields

We show that the elements of the ring of integers of real quadratic fields which are sums of integral squares are in fact sums of distinct squares, provided their norm is large enough.     

Sums of squares of regular functions on real algebraic varieties

Let be an affine algebraic variety over (or any other real closed field ). We ask when it is true that every positive semidefinite (psd) polynomial function on is a sum of squares (sos). We show that for the answer is always negative if has a real point. Also, if is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same holds if is a smooth surface with only real divisors at infinity. The ``compact' case is harder. We completely settle the case of smooth curves of genus : If such a curve has a complex point at infinity, then every psd function is sos, provided the field is archimedean. If is not archimedean, there are counter-examples of genus .

     

Sums of squares in octonion algebras

Sums of squares in composition algebras are investigated using methods from the theory of quadratic forms. For any integer octonion algebras of level and of level are constructed.

     

Sums of even powers in Archimedean rings

This note is devoted to the study of totally archimedean rings of regular functions. We extend Schmüdgen's theorem to this class of rings. Moreover, we show that, in such rings, every totally positive element is a sum of even powers of totally positive elements, and hence is a sum of even powers of units. Received: 21 October 1999; in final form: 10 November 2000 / Published online: 18 January 2002     

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.     

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.