We show that there are many (compact) convex semi-algebraic sets in euclidean space that do not have a semidefinite representation. This gives a negative answer to a question by Nemirovski, resp. it shows that the Helton-Nie conjecture is false.

Hilbert proved that a non-negative real quartic form f(x, y, z) is the sum of three squares of quadratic forms. We give a new proof which shows that if the plane curve Q defined by f is smooth, then f has exactly 8 such representations, up to equivalence. They correspond to those real 2-torsion points of the Jacobian of Q which are not represented by a conjugation-invariant divisor on Q.

Given an affine algebraic variety V over ℝ with real points V(ℝ) compact and a non-negative polynomial function f∈ℝ[V] with finitely many real zeros, we establish a local-global criterion for f to be a sum of squares in ℝ[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(ℝ) 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, Schmüdgen’s solution of the K-moment problem for compact K is extended, for dim (K)=1, to the case when K is virtually compact.

Given polynomials h1, . . . , hr ∈ R[x1, . . . , xn], we study the complexity of the problem of representing polynomials in the form f = s0 + s1h1 + · · ·+ srhr (∗) with sums of squares si. Let M be the cone of all f which admit such a representation. The problem is said to be stable if there exists a function φ : N → N such that every f ∈ M has a representation (∗) with deg(si) ≤ φ(deg(f)). The main result says that if the set K = {h1 ≥ 0, . . . , hr ≥ 0} has dimension ≥ 2 and the sequence h1, . . . , hr has the moment property (MP), then the problem is not stable. In particular, this includes the case when K is compact, dim(K) ≥ 2 and the cone M is multiplicatively closed.

We establish a duality in the cohomology of arbitrary tori over smooth but not necessarily projective curves over a p-adic field. This generalises Lichtenbaum–Tate duality between the Picard group and the Brauer group of a smooth projective curve.

The notions of totally indeflnite and weakly isotropic algebras with involution are introduced and a proof is given of the fact that a fleld satisfles the Efiective Diagonalization Property (ED) if and only if it satisfles the following weak Hasse principle: every totally indeflnite central simple algebra with involution of the flrst kind over the given fleld is weakly isotropic. This generalizes a known result from quadratic form theory.

Let k be a perfect field with cd k(i) less than or equal to 1. We say that a k-variety X satisfies the Hasse principle if either X(k(xi)) = empty set for some real closure k(xi) of k, or if X(k) not equal empty set. Let G be a connected linear group over k. We prove that every homogeneous space under G, defined over k, satisfies the Hasse principle. This confirms a conjecture of Colliot-Thelelene. In particular, the restriction map H-1(k, G) --> Pi H-1(k(xi), G) is injective, where the product is taken over the real closures of k. We also determine the image of this map and thereby express H-1(k, G) completely in terms of the orderings of k. As applications of these results we prove several approximation theorems for homogeneous spaces (weak approximation, component approximation).

This book makes a systematic study of the relations between the étale cohomology of a scheme and the orderings of its residue fields. A major result is that in high degrees, étale cohomology is cohomo

Definition 1. Ist (M, S) eine (partiell) geordnete Menge und X ⊆ M eine Teilmenge, so heißt X konvex in M, wenn für alle x, y, z ∈ M gilt: $x \leqslant z \leqslant y\,und\,x,y \in X \Rightarrow z \in X.$