A Generalization of the Milnor Number.
On the bifurcation set of complex polynomial with isolated singularities at infinity
Lipschitz stratification of real analytic sets
A note on singularities at infinity of complex polynomials
Let f be a complex polynomial. We relate the behaviour of f “at infinity” to the sheaf of vanishing cycles of the family of projective closures of fibres of f. We show that the absence of such cycles: (i) is equivalent to a condition on the asymptotic behaviour of gradient of f known as Malgrange’s Condition, (ii) implies the -triviality of f. If the support of sheaf of vanishing cycles of is a finite set, then it detects precisely the change of the topology of the fibres of f. Moreover, in...
Lipschitz properties of semi-analytic sets
The existence of Lipschitz stratification, in the sense of Mostowski, for compact semi-analytic sets is proved. (This stratification ensures the constance of the Lipschitz type along each stratum). The proof is independent of the complex case, considered by Mostowski, and gives also some other Lipschitz properties of semi-analytic sets.
Lipschitz stratification of subanalytic sets
Bi-Lipschitz trivialization of the distance function to a stratum of a stratification
Given a Lipschitz stratification 𝒳 that additionally satisfies condition (δ) of Bekka-Trotman (for instance any Lipschitz stratification of a subanalytic set), we show that for every stratum N of 𝒳 the distance function to N is locally bi-Lipschitz trivial along N. The trivialization is obtained by integration of a Lipschitz vector field.
Algebraically constructible functions and signs of polynomials.
Periodic solutions near an equilibrium of a differential equation with a first integral
On the Euler characteristic of fibres of real polynomial maps
Let Y be a real algebraic subset of and be a polynomial map. We show that there exist real polynomial functions on such that the Euler characteristic of fibres of is the sum of signs of .
Algebraically constructible functions
Motivic-type invariants of blow-analytic equivalence
To a given analytic function germ , we associate zeta functions , , defined analogously to the motivic zeta functions of Denef and Loeser. We show that our zeta functions are rational and that they are invariants of the blow-analytic equivalence in the sense of Kuo. Then we use them together with the Fukui invariant to classify the blow-analytic equivalence classes of Brieskorn polynomials of two variables. Except special series of singularities our method classifies as well the blow-analytic...
Invariants of bi-Lipschitz equivalence of real analytic functions
We construct an invariant of the bi-Lipschitz equivalence of analytic function germs (ℝⁿ,0) → (ℝ,0) that varies continuously in many analytic families. This shows that the bi-Lipschitz equivalence of analytic function germs admits continuous moduli. For a germ f the invariant is given in terms of the leading coefficients of the asymptotic expansions of f along the sets where the size of |x| |grad f(x)| is comparable to the size of |f(x)|.
Proof of the gradient conjecture of R. Thom.
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