Victoria Powers

An algorithm for sums of squares of real polynomials

Abstract We present an algorithm to determine if a real polynomial is a sum of squares (of polynomials), and to find an explicit representation if it is a sum of squares. This algorithm uses the fact that a sum of squares representation of a real polynomial corresponds to a real, symmetric, positive semi-definite matrix whose entries satisfy certain linear equations.

A new bound for Pólya's Theorem with applications to polynomials positive on polyhedra

Let R[X]≔R[x1,…,xn] and let and Δn denote the simplex {(x1,…,xn)|xi≥0,∑ixi=1}. Polyas Theorem says that if f∈R[X] is homogeneous and positive on Δn, then for sufficiently large N all of the coefficients of (x1+⋯+xn)Nf(x1,⋯,xn) are positive. We give an explicit bound for N and an application to some special representations of polynomials positive on polyhedra. In particular, we give a bound for the degree of a representation of a polynomial positive on a convex polyhedron as a positive linear combination of products of the linear polynomials defining the polyhedron.

Polynomials that are positive on an interval

This paper discusses representations of polynomials that are positive on intervals of the real line. An elementary and constructive proof of the following is given: If h(x), p(x) ∈ R[x] such that {α ∈ R | h(α) ≥ 0} = [−1, 1] and p(x) > 0 on [−1, 1], then there exist sums of squares s(x), t(x) ∈ R[x] such that p(x) = s(x) + t(x)h(x). Explicit degree bounds for s and t are given, in terms of the degrees of p and h and the location of the roots of p. This is a special case of Schmüdgen’s Theorem, and extends classical results on representations of polynomials positive on a compact interval. Polynomials positive on the non-compact interval [0,∞) are also considered.

Holomorphy rings and higher level orders on skew fields

The intimate connections which exist between higher level orders, sums of 2n th powers, valuation theory, and real-valued places on a field have been explored extensively. Craven initiated the study of similar notions for skew fields in [4]. In [12] we pursued these ideas, but the best results required a 2-primary assumption. This paper extends the work of [12]; in particular, we prove an “Artin-Schreier” theorem for preordered skew fields, namely that a preorder T in a skew field is the intersection of the kernels of all the signatures that contain T. This allows us to obtain the results of [12] without the 2-primary assumption. In particular, we have that preordered skew fields give rise to spaces of signatures, Mulcahy’s abstract setting for the theory of higher level forms and reduced Witt rings. Our main tools are valuation theory and the Kadison-Dubois representation theorem for partially ordered rings. As in the field case, we make use of the ring A, associated to a preorder T, which is the intersection of certain valuation rings. Along the way to our main theorem we prove some facts about noncommutative Priifer rings, and we show that A, is a Priifer ring. While our study of preordered skew fields follows along lines which are familiar from the field context, there are significant differences between the two theories. For example, there are skew fields which have no orders, but admit valuation rings with ordered residue fields.

A quantitative Pólya's Theorem with corner zeros

Pólyas Theorem says that if <i>p</i> is a homogeneous polynomial in <i>n</i> variables which is positive on the standard <i>n</i>-simplex, and <i>F</i> is the sum of the variables, then for a sufficiently large exponent <i>N, F^N * p</i> has positive coefficients. Pólyas Theorem has had many applications in both pure and applied mathematics; for example it provides a certificate for the positivity of <i>p</i> on the simplex. The authors have previously given an explicit bound on <i>N</i>, determined by the data of <i>p</i>; namely, the degree, the size of the coefficients and the minimum value of <i>p</i> on the simplex. In this paper, we extend this quantitative Pólyas Theorem to non-negative polynomials which are allowed to have simple zeros at the corners of the simplex.

Higher level orders on noncommutative rings

Abstract Becker introduced higher level orders on a field, a generalization of the notion of an order on a field. Higher level orders have since been defined for skew fields and commutative rings. We extend the notion of higher level orders to noncommutative rings and generalize some of the theory for commutative rings to the noncommutative setting. We obtain more results for a particular class of ring, those which modulo a prime ideal are embeddable in a skew field. Finally, we define the real spectrum of a noncommutative ring and obtain some topological results concerning the real spectrum.

Valuations and higher level orders in commutative rings

Valuation theory is one of the main tools for studying higher level orders and the reduced theory of forms over fields, see, for example [BR]. In [MW], the theory of higher level orders and reduced forms was generalized to rings with many units and many of the results for fields carried over to this setting. While it seems desirable to extend these results further, the techniques used for rings with many units will not work for general commutative rings. At the same time, there is a general theory of valuations in commutative rings (see [LM], [M], and [G]), which in [Ma] was used to study orders and the reduced theory of quadratic forms over general commutative rings. Thus it seems natural to ask if the connections between valuations and higher level orders in fields exist in commutative rings. In this paper we use valuation theory to study the space of orders and the reduced Witt ring relative to a higher level preorder in a commutative ring. As in [Ma], we first localize our ring at a multiplicative set, without changing the space of orders, in order to make the valuation theory work better. This is a standard idea from real algebraic geometry. Remarkably, many of the notions, methods, and results for fields carry over to this new setting. We define compatiblity between valuations and orders and preorders, and the ring A(T ) associated to a preorder T , which turns out to be Prufer ring as in the field case. We define the relation of dependency on the set of valuations associated to a preorder and we use this to prove a decomposition theorem for the space of orders. We can then apply this to show that, under a certain finiteness condition, the space of orders is equivalent to the space of orders of a preordered field.

Ordered Algebraic Structures and Related Topics

The present survey article has two aims: - To provide an intuitive and accessible introduction to the theory of the field of surreal numbers with exponential and logarithmic functions. - To give an overview of some of the recent achievements. In particular, the field of surreal numbers carries a derivation which turns it into a universal domain for Hardy fields.