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A note on singularities at infinity of complex polynomials

Adam Parusiński — 1997

Banach Center Publications

Let f be a complex polynomial. We relate the behaviour of f “at infinity” to the sheaf of vanishing cycles of the family f ¯ 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 C -triviality of f. If the support of sheaf of vanishing cycles of f ¯ 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

Adam Parusiński — 1988

Annales de l'institut Fourier

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.

Motivic-type invariants of blow-analytic equivalence

Satoshi KoikeAdam Parusiński — 2003

Annales de l'Institut Fourier

To a given analytic function germ f : ( d , 0 ) ( , 0 ) , we associate zeta functions Z f , + , Z f , - [ [ T ] ] , 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

Jean-Pierre HenryAdam Parusiński — 2004

Banach Center Publications

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)|.

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