We present a simple constructive proof of the existence of distinguished sum of squares representations for polynomials nonnegative over finite sets described by polynomial equalities and inequalities. A degree bound is directly obtained, as the cardinality of the support of the summands equals the number of points in the variety. Only basic results from commutative algebra are used in the construction.

We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free ∗-algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature.

Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates can be found by sophisticated numerical methods such as semidefinite programming, yet the actual verification of the resulting proof is straightforward even in a highly foundational theorem prover. We will describe our experience with an implementation in HOL Light, noting some successes as well as difficulties. We also describe a new approach to the univariate case that can handle some otherwise difficult examples. 1 Verifying nonlinear formulas over the reals Over the real numbers, there are algorithms that can in principle perform quantifier elimination from arbitrary first-order formulas built up using addition, multiplication and the usual equality and inequality predicates. A classic example of such a quantifier elimination equivalence is the criterion for a quadratic equation to have a real root: ∀a b c. (∃x. ax 2 + bx + c = 0) ⇔ a = 0 ∧ (b = 0 ⇒ c = 0) ∨ a � = 0 ∧ b 2 ≥ 4ac

We consider the problem of minimizing a form on the standard simplex [equivalently, the problem of minimizing an even form on the unit sphere]. Converging hierarchies of approximations for this problem can be constructed, that are based, respectively, on results by Schmudgen-Putinar and by Polya about representations of positive polynomials in terms of sums of squares. We show that the two approaches yield, in fact, the same approximations. The same type of argument also permits to establish some representation results a la Polya for positive polynomials on semi-algebraic cones.

Abstract. An extension of the Kadison-Dubois representation theorem is proved. This extends both the classical version [3] and the preordering version given by Jacobi in [5]. It is then shown how this can be used to sharpen the results on representations of strictly positive polynomials given by Jacobi and Prestel in [6]. In [4] Dubois extends a result of Kadison on representation of archimedean partially ordered Banach algebras [7] to a more general class of rings called Stone rings. This is the so-called Kadison-Dubois theorem. 1 It is used in real algebra, e.g., by Becker [2], in his study of sums of 2n-th powers and the real holomorphy ring of a field. In [3] Becker and Schwartz give a short elementary proof in the commutative case. In his solution of the moment problem in [13], Schmüdgen gives a representation of polynomials strictly positive on a bounded basic closed semi-algebraic set in Rn.Putinar [12] gives a criterion for ‘linear ’ representations to exist. Jacobi and Prestel [6] show how Schmüdgen’s representation can be improved and determine exactly when the linear representations considered by Putinar are possible. Schmüdgen and Putinar use methods from functional analysis. In [14] Wörmann uses the Kadison-Dubois theorem to give a purely algebraic proof of Schmüdgen’s result. In [5] Jacobi proves a new variant of the Kadison-Dubois theorem and uses this to give an algebraic proof of Putinar’s criterion for linear representations. In the present paper we prove a general representation theorem for archimedean T-modules, T a weakly torsion preprime; see Theorem 2.3. This extends both the classical Kadison-Dubois theorem in [3] and the version given by Jacobi in [5]. In Section 3 it is explained how this can be used to sharpen the results on representations of strictly positive polynomials given in [6]; also see [10, Sect. 5.4].

Abstract. We consider the problem of determining the closure M of a quadratic module M in a commutative R-algebra with respect to the finest locally convex topology. This is of interest in deciding when the moment problem is solvable [26] [27] and in analyzing algorithms for polynomial optimization involving semidefinite programming [12]. The closure of a semiordering is also considered, and it is shown that the space YM consisting of all semiorderings lying over M plays an important role in understanding the closure of M. The result of Schmüdgen for preorderings in [27] is strengthened and extended to quadratic modules. The extended result is used to construct an example of a non-archimedean quadratic module describing a compact semialgebraic set that has the strong moment property. The same result is used to obtain a recursive description of M which is valid in many cases. In Section 1 we consider the general relationship between the closure C and the sequential closure C ‡ of a subset C of a real vector space V in the finest locally convex topology. We are mainly interested in the case where C is a cone in V. We consider cones with non-empty interior and cones satisfying C ∪ −C = V. In Section 2 we begin our investigation of the closure M of a quadratic module M of a commutative R-algebra A; the focus is on finitely generated quadratic modules in finitely generated algebras. The closure of a semiordering Q of A is also considered, and it is shown that the space YM consisting of all semiorderings of A lying over M plays an important role in understanding the closure of M; see Propositions 2.2, 2.3 and 2.4. The result of Schmüdgen for preorderings in [27] is strengthened and extended to quadratic modules; see Theorem 2.8. In Section 3 we consider the case of quadratic modules that describe compact semialgebraic sets. We use Theorem 2.8 to deduce various results; see Theorems 3.1 and 3.4; and also to construct an example where KM is compact, M satisfies the strong moment property (SMP), but M is not archimedean; see Example 3.7.

During the last 10 years there have been several results on the representation of real polynomials, positive on some semi-algebraic subset of R n. These results started with a solution of the moment problem by Schmüdgen for corresponding sets. Later Wörmann realized that the same results could be obtained by the so-called “Kadison-Dubois ” Representation Theorem. The aim of our talk is to present this representation theorem together with its history, and to discuss its implication to the representation of positive polynomials. Also recent improvements of both topics by T. Jacobi and the author will be included. 1 The real representation theorem. Roughly speaking, the purpose of a representation theorem is, to start with awell-known mathematical structure, generalize (axiomatize) it and try to ‘represent ’ the generalized structures in terms of the old one. In our case, the ring that is assumed to be well-known is the ring C(X, R) of continuous real-valued functions on a compact hausdorff space X. This ring carries a natural partial order given by Thus the set f ≤ g iff f(x) ≤ g(x) for all x ∈ X. T0 = {f ∈ C(X, R) | f ≥ 0onX} is a preordering of the ring C = C(X, R), i.e., we have

Following the classical approach of Pólya-Schur theory [14] we initiate in this paper the study of linear operators acting on R[x] and preserving either the set of positive univariate polynomials or similar sets of non-negative and elliptic polynomials.

. By Schmudgen's Theorem, polynomials f 2 R[X1 ; : : : ; Xn ] strictly positive on a bounded basic semialgebraic subset of R n , admit a certain representation involving sums of squares oe i from R[X1 ; : : : ; Xn ]. We show the existence of effective bounds on the degrees of the oe i by proving first a suitable version of Schmudgen's Theorem over nonarchimedean real closed fields, and then applying the Compactness and Completeness Theorem from Model Theory. 1. Statement of the Results In this paper we deal with `effectivity problems' in connection with the following theorem proved by K. Schmudgen in [Sch]. Let R[X] = R[X 1 ; : : : ; Xn ] be the ring of real polynomials in X 1 ; : : : ; Xn . For h 1 ; : : : ; hm 2 R[X], the set S(h) = S(h 1 ; : : : ; hm ) = fa 2 R n j h 1 (a) 0; : : : ; hm (a) 0g is a basic closed semi-algebraic subset of R n . Schmudgen's Theorem states that if S(h) is bounded, then every f 2 R[X] that is strictly positive on S(h) admits a represent...

Pólya proved that if a real, homogeneous polynomial is positive on the nonnegative orthant (except at the origin), then it is the quotient of two homogeneous polynomials with no negative coefficients. We generalize this from polynomials to signomials with arbitrary rational exponents; we also show that Pólya’s theorem does not generalize to arbitrary signomials (i.e., with irrational (real) exponents). 1