Thomas C. Craven

Composition Theorems, Multiplier Sequences and Complex Zero Decreasing Sequences

An important chapter in the theory of distribution of zeros of polynomials and transcendental entire functions pertains to the study of linear operators acting on entire functions. This article surveys some recent developments (as well as some classical results) involving some specific classes of linear operators called multiplier sequences and complex zero decreasing sequences. This expository article consists of four parts: Open problems and background information, Composition theorems (Section 2), Multiplier sequences and the Laguerre-Polya class (Section 3) and Complex zero decreasing sequences (Section 4). A number of open problems and questions are also included.

Witt rings and orderings of skew fields

While ordered skew fields have been known for a long time, they have received very little attention. One purpose of this paper is to show that studying orderings and associated structures is one way of obtaining information about skew fields which are infinite dimensional over their centers. All of our work is restricted to skew fields of characteristic not equal to 2. Of course, ordered skew fields have characteristic zero since they induce orderings on their centers. The first example of a noncommutative ordered field appears to be due to D. Hilbert in “Grundlagen der Geometrie” 115,

Characterizing reduced Witt rings of fields

331. The example follows a proof of the fact that if a skew lield D has an archimedean ordering (every positive element of D is less than some rational integer), then D is commutative [15,

The Boolean space of orderings of a field

321. Another early result of a general nature was proved by A. A. Albert [I 1, based on earlier work of L. Dickson. Albert showed that an ordered skew field which is algebraic over its center must be commutative. We shall generalize this result in the next section. In 1952, T. Szele [28 ] extended to skew fields the basic results of Artin and Schreier on sums of squares [2,3] ( sums of products of squares for skew fields) and a criterion for extension of orderings due to Serre 1271. The second and third sections of our paper are devoted to extending these results and others to “orderings of higher level” in skew fields. Orderings of higher level were recently defined by E. Becker in [4] for commutative fields and appear to be of great importance in extending results for quadratic forms to forms of higher degree. In particular, 2”th powers take the place of squares in the Artin-Schreier theory. In fact, in (5 1 he has extended this to arbitrary even powers, but the valuation theory needed exceeds that which we have developed in Section 3 for skew fields. Section 2 covers the extension to skew

A Sufficient condition for strict total positivity of a matrix

Let IV(F) denote the Mitt ring of nondegenerate symmetric bilinear forms over a field F. In this paper wc shall be concerned only with formally real fields, for which we write Wr,,l(F) ~mm W(F)/Wil W(F) for the reduced R’itt ring. In [13, 141 the rings W(F) and iTred are shown to be special cases of absfrart lWtt rirqs and a great deal of the ring structure is developed in this setting. In [6] it is shown that not all of these abstract Wtt rings can be FVitt rings of fields and more examples are given in [7]. In this paper we shall show precisel! which of the torsion fret abstract Witt rings (subject to a certain finiteness restriction) can be reduced Witt rings of fields. In Section 2 we give an inductive construction of all reduced Witt rings of fields with only finitely many places into the real numbers R. This construction provides a powerful tool for proving ring-theoretic facts about reduced \Vitt rings. We apply this construction in Section 3 to obtain an explicit description of the structure of these rings in terms of the real places on any field whose reduced Witt ring is isomorphic to the given ring. In Section 4 we look at another application of the indutcive construction. \

Approximation properties for orderings on *-fields

‘e pro\-e the following conjecture in the case that F is a field with only finitely many places into [w: If cp E W(F) ma s into PF, for each real closure p. F, of F, then q~ is in WJF) 1 IIF, w ere IF denotes the maximal ideal of all even h dimensional forms o\-cr F and W,(F) d enotes the torsion subgroup of W(F). Before we begin our inductive construction, we shall need some definitions and notation. As in [ 131, wc shall write X(F) or X( W,,,l(F)) for the Boolean space of orderings of a field F, and we shall think of Wred(F) as a subring of ‘&(X(F), Z), the ring of all continuous functions from -Y(F) to Z, where Z has the discrete topology. Recall that the topology of X(F) is induced by the Harrison subbasis, which consists of all sets of the form

The map of the Witt ring of a domain into the Witt ring of its field of fractions

It has been pointed out by Knebusch, Rosenberg and Ware that the set X of all orderings on a formally real field can be topologized to make a Boolean space (compact, Hausdorff and totally disconnected). They have called the sets of orderings W(a) = {< in Xia < 0} the Harrison subbasis of X. This subbasis is closed under symmetric difference and complementation. In this paper it is proved that, given any Boolean space X, there exists a formally real field F such that X is homeomorphic to the space of orderings on F. Also, an example is given of a Boolean space and a basis of clopen sets closed under symmetric difference and complementation which cannot be the Harrison subbasis of any formally

Witt groups of hermitian forms over ∗-fields

We establish a sufficient condition for strict total positivity of a matrix In particular, we show that if the (positive) elements of a square matrix grow sufficiently fast as their distance from the diagonal of the matrix increases, then the matrix is strictly totally positive.

Zero-diminishing linear transformations

The goal of this paper is to extend the main theorems on approximation properties of the topological space of orderings from formally real fields to skew fields with an involution *. To accomplish this, the concept of *-semiordering is developed and new theorems are obtained for lifting orderings from the residue class field of a real valuation.

Existence of SAP extension fields

Let R be an integral domain with field of fractions K. This paper studies the kernel of the map W(R) _ W(K), where W is the Witt ring functor. In case R is regular and noetherian, it is shown that the kernel is a nilideal. The kernel is zero if R is a complete regular local noetherian ring with 2 a unit. Examples are given to show that the regularity assumptions are needed.