On nilpotent filiform Lie algebras of dimension eight

The aim of this paper is to determine both the Zariski constructible set of characteristically nilpotent filiform Lie algebrasgof dimension8and that of the set of nilpotent filiform Lie algebras whose group of automorphisms consists of unipotent automorphisms, in the variety of filiform Lie algebras of dimension8overC.

Open questions in the theory of spaces of orderings

Spaces of orderings provide an abstract framework in which to study spaces of orderings of formally real fields. Spaces of orderings of finite chain length are well understood [9, 11]. The Isotropy Theorem [11] and the extension of the Isotropy Theorem given in [13] are the main tools for reducing questions to the finite case, and these are quite effective. At the same time, there are many questions which do not appear to reduce in this way. In this paper we consider four such questions, for a space of orderings (X, G).1. Is it true that every positive primitive formula P(a) with parameters a in G which holds in every finite subspace of (X, G) necessarily holds in (X, G)?2. If f: X → ℤ is continuous and Σx∈Vf(x) ≡ 0 mod ∣V∣ holds for all fans V in X with ∣V∣ ≤ 2n, does there exist a form ϕ with entries in G such that mod Cont(X, 2nℤ)?3. Is it true that Cont(X, 2nℤ) ∩ Witt(X, G) = In(X, G), where I(X, G) denotes the fundamental ideal?4. Is the separating depth of a constructible set C in X necessarily bounded by the stability index of (X, G)?The unexplained terminology and notation is explained later in the main body of the paper. In a certain sense Question 1 is the main question. At the same time, Questions 2, 3 and 4 are of considerable interest, both from the point of view of quadratic form theory and from the point of view of real algebraic geometry.