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On some noetherian rings of C ∞ germs on a real closed field

Abdelhafed Elkhadiri (2011)

Annales Polonici Mathematici

Let R be a real closed field, and denote by R , n the ring of germs, at the origin of Rⁿ, of C ∞ functions in a neighborhood of 0 ∈ Rⁿ. For each n ∈ ℕ, we construct a quasianalytic subring R , n ⊂ R , n with some natural properties. We prove that, for each n ∈ ℕ, R , n is a noetherian ring and if R = ℝ (the field of real numbers), then ℝ , n = ₙ , where ₙ is the ring of germs, at the origin of ℝⁿ, of real analytic functions. Finally, we prove the Real Nullstellensatz and solve Hilbert’s 17th Problem for the ring R , n .

On special partial types and weak canonical bases in simple theories

Ziv Shami (2013)

Fundamenta Mathematicae

We define the notion of a weak canonical base for a partial type in a simple theory. We prove that members of a certain family of partial types, which we call special partial types, admit a weak canonical base; this family properly contains the family of amalgamation bases.

On the Boffa alternative

B. Bajorska, O. Macedońska (2001)

Colloquium Mathematicae

Let G* denote a nonprincipal ultrapower of a group G. In 1986 M.~Boffa posed a question equivalent to the following one: if G does not satisfy a positive law, does G* contain a free nonabelian subsemigroup? We give the affirmative answer to this question in the large class of groups containing all residually finite and all soluble groups, in fact, all groups considered in traditional textbooks on group theory.

On the Cantor-Bendixson rank of metabelian groups

Yves Cornulier (2011)

Annales de l’institut Fourier

We study the Cantor-Bendixson rank of metabelian and virtually metabelian groups in the space of marked groups, and in particular, we exhibit a sequence ( G n ) of 2-generated, finitely presented, virtually metabelian groups of Cantor-Bendixson rank  ω n .

On the Euler characteristic of the links of a set determined by smooth definable functions

Krzysztof Jan Nowak (2008)

Annales Polonici Mathematici

The purpose of this paper is to carry over to the o-minimal settings some results about the Euler characteristic of algebraic and analytic sets. Consider a polynomially bounded o-minimal structure on the field ℝ of reals. A ( C ∞ ) smooth definable function φ: U → ℝ on an open set U in ℝⁿ determines two closed subsets W := u ∈ U: φ(u) ≤ 0, Z := u ∈ U: φ(u) = 0. We shall investigate the links of the sets W and Z at the points u ∈ U, which are well defined up to a definable homeomorphism. It is proven...

On the existence of super-decomposable pure-injective modules over strongly simply connected algebras of non-polynomial growth

Stanisław Kasjan, Grzegorz Pastuszak (2014)

Colloquium Mathematicae

Assume that k is a field of characteristic different from 2. We show that if Γ is a strongly simply connected k-algebra of non-polynomial growth, then there exists a special family of pointed Γ-modules, called an independent pair of dense chains of pointed modules. Then it follows by a result of Ziegler that Γ admits a super-decomposable pure-injective module if k is a countable field.

On the implicit function theorem in o-minimal structures

Zofia Ambroży, Wiesław Pawłucki (2015)

Banach Center Publications

A local-global version of the implicit function theorem in o-minimal structures and a generalization of the theorem of Wilkie on covering open sets by open cells are proven.

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