A note on the Łojasiewicz exponent of non-degenerate isolated hypersurface singularities

Brzostowski, Szymon

doi:10.18778/8142-814-9.04

Cited by 5 publications

(3 citation statements)

References 15 publications

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“…As an immediate consequence of Theorem 2.1, we get the following strengthening of the main result of [Brz19] (see Theorem 3.1 in the next section).…”

Section: Commentssupporting

confidence: 57%

The Łojasiewicz exponent of nondegenerate surface singularities

Oleksik¹

2023

Can. Math. Bull.

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Let f be an isolated singularity at the origin of $\mathbb {C}^n$ . One of many invariants that can be associated with f is its Łojasiewicz exponent $\mathcal {L}_0 (f)$ , which measures, to some extent, the topology of f. We give, for generic surface singularities f, an effective formula for $\mathcal {L}_0 (f)$ in terms of the Newton polyhedron of f. This is a realization of one of Arnold’s postulates.

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“…As an immediate consequence of Theorem 2.1, we get the following strengthening of the main result of [Brz19] (see Theorem 3.1 in the next section).…”

Section: Commentssupporting

confidence: 57%

The Łojasiewicz exponent of nondegenerate surface singularities

Oleksik¹

2023

Can. Math. Bull.

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“…For isolated singularity case, this does not happen. In fact, Brzostowski proved the Lojasiewicz exponent η 0 (f ) of the Lojasiewicz inequality of type ( 2) is constant on the moduli space of functions with fixed Newton boundary and an isolated singularity at the origin (Theorem 1, [4]). On the other hand, θ 0 (f ) and η 0 (f ) are related by θ 0 (f ) = η 0 (f )/(1 + η 0 (f )) by Teissier [23].…”

Section: Examples Of the Estimation Of θ 0 (F )mentioning

confidence: 99%

Łojasiewicz exponents of a certain analytic functions

Oka¹

2020

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We consider the exponent of Lojasiewicz inequality ∂ f (z) ≥ c|f (z| θ for two classes of analytic functions and we will give an explicit estimation for θ. First we consider certain non-degenerate functions which is not convenient. In §3.4, we give an example of a polynomial for which θ0(f ) is not constant on the moduli space and in §3.5, we show that the behaviors of the Lojasiewicz exponents is not similar as the Milnor numbers by an example.In the last section ( §4), we give also an estimation for product functions f (z) = f1(z) • • • f k (z) associated to a family of a certain convenient non-degenerate complete intersection varieties. In either class, the singularity is not isolated. We will give explicit estimations of the Lojasiewicz exponent θ0(f ) using combinatorial data of the Newton boundary of f . We generalize this estimation for non-reduced function g2000 Mathematics Subject Classification. 32S05.

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“…2.15] to the n-dimensional case. Since the Łojasiewicz exponent is one and the same for isolated non-degenerate singularities with a given Newton polyhedron (see [2]), there "only" remains the question about a formula for the exponent in terms of the Newton polyhedron (cf. Remark 5).…”

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confidence: 99%