Note on convergence and orders of vanishing along curves.
Note on the degree of Cº-sufficiency of plane curves.
Let f be a germ of plane curve, we define the δ-degree of sufficiency of f to be the smallest integer r such that for anuy germ g such that j(r) f = j(r) g then there is a set of disjoint annuli in S3 whose boundaries consist of a component of the link of f and a component of the link of g. We establish a formula for the δ-degree of sufficiency in terms of link invariants of plane curves singularities and, as a consequence of this formula, we obtain that the δ-degree of sufficiency is equal to the...
Nullstellensatz and cycles of zeroes of holomorphic mappings
The local Nullstellensatz exponent for holomorphic mappings via intersection theory for the cases of isolated and quasi-complete intersection is considered.
O-minimal version of Whitney's extension theorem
This is a generalized and improved version of our earlier article [Studia Math. 124 (1997)] on the Whitney extension theorem for subanalytic -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 -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 analytic models of synthetic differential geometry
On analytic sets and functions with given isolated singularities.
On arc-analytic functions definable by a Weierstrass system
This paper presents certain characterizations through blowing up of arc-analytic functions definable by a convergent Weierstrass system closed under complexification.
On Automorphisms of Complements of Analytic Subsets in Cn.
On gradient at infinity of semialgebraic functions
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 such that |x|·|∇f| and are separated at infinity. If c is a regular value and , 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
We prove the o-minimal generalization of the Łojasiewicz inequality , with , in a neighborhood of , where is real analytic at and . 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 hypersurface singularities which are stems
On improper isolated intersection in complex analytic geometry
On irreducible components of a Weierstrass-type variety
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 irregular links at infinity of algebraic plane curves.
On microlocal -function
On quasi-free Hilbert modules.
On roots of polynomials with power series coefficients
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
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 semi-analytic and subanalytic geometry
On simple germs with non-isolated singularities.