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O-minimal version of Whitney's extension theorem

Krzysztof Kurdyka, Wiesław Pawłucki (2014)

Studia Mathematica

This is a generalized and improved version of our earlier article [Studia Math. 124 (1997)] on the Whitney extension theorem for subanalytic p -Whitney fields (with p finite). In this new version we consider Whitney fields definable in an arbitrary o-minimal structure on any real closed field R and obtain an extension which is a p -function definable in the same o-minimal structure. The Whitney fields that we consider are defined on any locally closed definable subset of Rⁿ. In such a way, a local...

On gradient at infinity of semialgebraic functions

Didier D'Acunto, Vincent Grandjean (2005)

Annales Polonici Mathematici

Let f: ℝⁿ → ℝ be a C² semialgebraic function and let c be an asymptotic critical value of f. We prove that there exists a smallest rational number ϱ c ≤ 1 such that |x|·|∇f| and | f ( x ) - c | ϱ c are separated at infinity. If c is a regular value and ϱ c < 1 , then f is a locally trivial fibration over c, and the trivialisation is realised by the flow of the gradient field of f.

On gradients of functions definable in o-minimal structures

Krzysztof Kurdyka (1998)

Annales de l'institut Fourier

We prove the o-minimal generalization of the Łojasiewicz inequality ∥ grad f ∥ ≥ | f | α , with α &lt; 1 , in a neighborhood of a , where f is real analytic at a and f ( a ) = 0 . We deduce, as in the analytic case, that trajectories of the gradient of a function definable in an o-minimal structure are of uniformly bounded length. We obtain also that the gradient flow gives a retraction onto levels of such functions.

On irreducible components of a Weierstrass-type variety

Romuald A. Janik (1997)

Annales Polonici Mathematici

We give a characterization of the irreducible components of a Weierstrass-type (W-type) analytic (resp. algebraic, Nash) variety in terms of the orbits of a Galois group associated in a natural way to this variety. Since every irreducible variety of pure dimension is (locally) a component of a W-type variety, this description may be applied to any such variety.

On roots of polynomials with power series coefficients

Rafał Pierzchała (2003)

Annales Polonici Mathematici

We give a deepened version of a lemma of Gabrielov and then use it to prove the following fact: if h ∈ 𝕂[[X]] (𝕂 = ℝ or ℂ) is a root of a non-zero polynomial with convergent power series coefficients, then h is convergent.

On semialgebraic points of definable sets

Artur Piękosz (1998)

Banach Center Publications

We prove that the semialgebraic, algebraic, and algebraic nonsingular points of a definable set in o-minimal structure with analytic cell decomposition are definable. Moreover, the operation of taking semialgebraic points is idempotent and the degree of complexity of semialgebraic points is bounded.

On some global semianalytic sets

Abdelhafed Elkhadiri (2013)

Annales de l’institut Fourier

We give some structures without quantifier elimination but in which the closure, and hence the interior and the boundary, of a quantifier free definable set is also a quantifier free definable set.

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 the Bernstein-Walsh-Siciak theorem

Rafał Pierzchała (2012)

Studia Mathematica

By the Oka-Weil theorem, each holomorphic function f in a neighbourhood of a compact polynomially convex set K ⊂ ℂ N can be approximated uniformly on K by complex polynomials. The famous Bernstein-Walsh-Siciak theorem specifies the Oka-Weil result: it states that the distance (in the supremum norm on K) of f to the space of complex polynomials of degree at most n tends to zero not slower than the sequence M(f)ρ(f)ⁿ for some M(f) > 0 and ρ(f) ∈ (0,1). The aim of this note is to deduce the uniform version,...

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