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On a problem concerning quasianalytic local rings

Hassan Sfouli — 2014

Annales Polonici Mathematici

Let (ₙ)ₙ be a quasianalytic differentiable system. Let m ∈ ℕ. We consider the following problem: let $f \in_{m}$ and f̂ be its Taylor series at $0 \in ℝ^{m}$ . Split the set $ℕ^{m}$ of exponents into two disjoint subsets A and B, $ℕ^{m} = A \cup B$ , and decompose the formal series f̂ into the sum of two formal series G and H, supported by A and B, respectively. Do there exist $g, h \in_{m}$ with Taylor series at zero G and H, respectively? The main result of this paper is the following: if we have a positive answer to the above problem for some m ≥ 2, then...

Weierstrass division theorem in quasianalytic local rings

Abdelhafed Elkhadiri; Hassan Sfouli — 2008

Studia Mathematica

The main result of this paper is the following: if the Weierstrass division theorem is valid in a quasianalytic differentiable system, then this system is contained in the system of analytic germs. This result has already been known for particular examples, such as the quasianalytic Denjoy-Carleman classes.

Weierstrass division theorem in definable $C^{\infty}$ germs in a polynomially bounded o-minimal structure

Abdelhafed Elkhadiri; Hassan Sfouli — 2006

Annales Polonici Mathematici

We give some examples of polynomially bounded o-minimal expansions of the ordered field of real numbers where the Weierstrass division theorem does not hold in the ring of germs, at the origin of ℝⁿ, of definable $C^{\infty}$ functions.

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On a problem concerning quasianalytic local rings

Weierstrass division theorem in quasianalytic local rings

Weierstrass division theorem in definable C ∞ germs in a polynomially bounded o-minimal structure

Weierstrass division theorem in definable $C^{\infty}$ germs in a polynomially bounded o-minimal structure