The Łojasiewicz exponent of semiquasihomogeneous singularities

Brzostowski, Szymon

doi:10.1112/blms/bdv058

Cited by 17 publications

(17 citation statements)

References 15 publications

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“…We consider a weighted homogeneous polynomial with isolated singularity at the origin. There are nice results by Abderrhmane [1] and Brzostowski [3]. I thank to Tadeusz Krasiński for informing me these papers.…”

Section: 72mentioning

confidence: 92%

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Łojasiewicz exponents of non-degenerate holomorohic and mixed functions

Oka¹

2018

Kodai Math. J.

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We consider Lojasiewicz inequalities for a non-degenerate holomorphic function with an isolated singularity at the origin. We give an explicit estimation of the Lojasiewicz exponent in a slightly weaker form than the assertion in Fukui [9]. For a weighted homogeneous polynomial, we give a better estimation in the form which is conjectured by [4] under under some condition (the Lojasiewicz non-degeneracy). We also introduce Lojasiewicz inequality for strongly non-degenerate mixed functions and generalize this estimation for mixed functions.2000 Mathematics Subject Classification. 14P05,32S55.

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Section: 72mentioning

confidence: 92%

“…By (3) and by the non-degeneray assumption, we have ord ∂f (z(t)) ≤ d − m(P ) (7) ord f j (z(t)) ≥ d(P, f j ) ≥ d − p j (8) ℓ 0 (C(t)) ≤ d − m(P ) m(P ) .…”

Section: C(t)mentioning

confidence: 99%

Łojasiewicz exponents of non-degenerate holomorohic and mixed functions

Oka¹

2018

Kodai Math. J.

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“…We recall that, by the main result of [12], if f : (C n , 0) → (C, 0) is a semi-weighted homogeneous function such that d w (f ) = d, then L 0 (∇f ) = d−w 0 w 0 , provided that d 2w i , for all i = 1, . .…”

Section: Mixed Lojasiewicz Exponents and Hickel Idealsmentioning

confidence: 99%

The sequence of mixed Łojasiewicz exponents associated to pairs of monomial ideals

Bivià-Ausina

2020

Journal of Algebra

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“…A. Parusiński called my attention to S. Brzostowski's result [2], which has independently proved the main theorem of this paper. But his proof is different from ours.…”

Section: The Maximal and Minimal Coordinatesmentioning

confidence: 99%

The Łojasiewicz Exponent for Weighted Homogeneous Polynomial With Isolated Singularity

Abderrahmane¹

2016

Glasgow Math. J.

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Abstract. The purpose of this paper is to give an explicit formula of the Lojasiewicz exponent of an isolated weighted homogeneous singularity in terms of its weights.Let f : (C n , 0) → (C, 0) be a holomorphic function with an isolated critical point at 0.It is well known(see [10]) that the Lojasiewicz exponent can be calculated by means of analytic pathswhere ord(φ) := inf i {ord(φ i )} for φ ∈ C{t} n . By definition, we put ord(0) = +∞.Lojasiewicz exponents have important applications in singularity theory, for instance, Teissier [22] showed that C 0 -sufficiency degree of f (i.e., the minimal integer r such that f is topologically equivalent to f + g for all g with ord(g)Despite deep research of experts in singularity theory, it is not proved yet that Lojasiewicz exponent L(f ) is a topological invariant of f (in contrast to the Milnor number). An interesting mathematical problem is to give formulas for L(f ) in terms of another invariants of f or an algorithm to compute it. In the two-dimensional case there are many explicit formulas for L(f ) in various terms (see The aim of this paper is to compute the Lojasiewicz exponent for the classes of weighted homogeneous isolated singularities in terms of the weights. In particular, we generalize a formula for L(f ) of Krasiński, Oleksik and P loski [8] for weighted homogeneous surface singularity. This was already announced by Tan, Yau and Zuo [21], but thier paper seems to have some gaps in the proof of proposition 3.4. We were motived by their papers. However, our considerations are based on other ideas. More precisely, we use the notion of weighted homogenous filtration introduced by Paunescu in [18], the geometric characterization of µ-constancy in [12,22] and the result of Varchenko [23], which described the µ-constant stratum of weighted homogeneous singularities in terms of the mixed Hodge structures.Moreover, we show that the Lojasiewicz exponent is invariant for all µ-constant deformation of weighted homogeneous singularity, which gives an affirmative partial answer to Teissier's conjecture [22].

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The Łojasiewicz exponent of semiquasihomogeneous singularities

Cited by 17 publications

References 15 publications

Łojasiewicz exponents of non-degenerate holomorohic and mixed functions

Łojasiewicz exponents of non-degenerate holomorohic and mixed functions

The sequence of mixed Łojasiewicz exponents associated to pairs of monomial ideals

The Łojasiewicz Exponent for Weighted Homogeneous Polynomial With Isolated Singularity

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