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Manifolds with a unique embedding

Zbigniew Jelonek (2009)

Colloquium Mathematicae

We show that if X, Y are smooth, compact k-dimensional submanifolds of ℝⁿ and 2k+2 ≤ n, then each diffeomorphism ϕ: X → Y can be extended to a diffeomorphism Φ: ℝⁿ → ℝⁿ which is tame (to be defined in this paper). Moreover, if X, Y are real analytic manifolds and the mapping ϕ is analytic, then we can choose Φ to be also analytic. We extend this result to some interesting categories of closed (not necessarily compact) subsets of ℝⁿ, namely, to the category of Nash submanifolds...

Matching local Witt invariants

Przemysław Koprowski (2005)

Acta Mathematica Universitatis Ostraviensis

The starting point of this note is the observation that the local condition used in the notion of a Hilbert-symbol equivalence and a quaternion-symbol equivalence — once it is expressed in terms of the Witt invariant — admits a natural generalisation. In this paper we show that for global function fields as well as the formally real function fields over a real closed field all the resulting equivalences coincide.

Michael's theorem for Lipschitz cells in o-minimal structures

Małgorzata Czapla, Wiesław Pawłucki (2016)

Annales Polonici Mathematici

A version of Michael's theorem for multivalued mappings definable in o-minimal structures with M-Lipschitz cell values (M a common constant) is proven. Uniform equi-LCⁿ property for such families of cells is checked. An example is given showing that the assumption about the common Lipschitz constant cannot be omitted.

Modified Nash triviality of a family of zero-sets of real polynomial mappings

Toshizumi Fukui, Satoshi Koike, Masahiro Shiota (1998)

Annales de l'institut Fourier

In this paper we introduce the notion of modified Nash triviality for a family of zero sets of real polynomial map-germs as a desirable one. We first give a Nash isotopy lemma which is a useful tool to show triviality.Then, using it, we prove two types of modified Nash triviality theorem and a finite classification theorem for this triviality. These theorems strengthen similar topological results.

Multi-Harnack smoothings of real plane branches

Pedro Daniel González Pérez, Jean-Jacques Risler (2010)

Annales scientifiques de l'École Normale Supérieure

Let Δ ⊂ 𝐑 2 be an integral convex polygon. G. Mikhalkin introduced the notion ofHarnack curves, a class of real algebraic curves, defined by polynomials supported on Δ and contained in the corresponding toric surface. He proved their existence, viaViro’s patchworkingmethod, and that the topological type of their real parts is unique (and determined by Δ ). This paper is concerned with the description of the analogous statement in the case of a smoothing of a real plane branch ( C , 0 ) . We introduce the class...

Multivariate moment problems : geometry and indeterminateness

Mihai Putinar, Claus Scheiderer (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

The most accurate determinateness criteria for the multivariate moment problem require the density of polynomials in a weighted Lebesgue space of a generic representing measure. We propose a relaxation of such a criterion to the approximation of a single function, and based on this condition we analyze the impact of the geometry of the support on the uniqueness of the representing measure. In particular we show that a multivariate moment sequence is determinate if its support has dimension one and...

Multivariate polynomial inequalities viapluripotential theory and subanalytic geometry methods

W. Pleśniak (2006)

Banach Center Publications

We give a state-of-the-art survey of investigations concerning multivariate polynomial inequalities. A satisfactory theory of such inequalities has been developed due to applications of both the Gabrielov-Hironaka-Łojasiewicz subanalytic geometry and pluripotential methods based on the complex Monge-Ampère operator. Such an approach permits one to study various inequalities for polynomials restricted not only to nice (nonpluripolar) compact subsets of ℝⁿ or ℂⁿ but also their versions for pieces...

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