Multiplicity and the Lojasiewicz exponent

Płoski, Arkadiusz

doi:10.4064/-20-1-353-364

Cited by 31 publications

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“…Moreover, if F is a regular sequence (i.e. n = m), then for generic direction ∈ P n−1 the exponent L 0 (F) is attained on the curve F −1 ( ) (see [15] or [16] for a different proof of this result). Another observation of this kind is the following In particular, if F = 0 and n = 2 then the local Łojasiewicz exponent of F is attained on one of the curves { f i = 0}.…”

Section: This Exponent Is Called the Local łOjasiewicz Exponent Of F mentioning

confidence: 99%

“…Example 3 A. Płoski in [16] proved that a rational number is equal to the Łojasiewicz exponent of a holomorphic mapping of C 2 if and only it appears in the sequence …”

Section: T)mentioning

confidence: 99%

“…Inspired by [9] Now, since it is an easy matter to actually find parametrizations giving equality in the above formula, we conclude by Theorem 7 that Question 2 Is our definition of the local Łojasiewicz exponent equivalent to Lejeune and Teissier's "integral closure definition" used in [3], or to Płoski's "characteristic polynomial definition" (cf. [16]), for every algebraically closed field K?…”

Section: T)mentioning

confidence: 99%

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The Łojasiewicz exponent over a field of arbitrary characteristic

Brzostowski

Rodak

2015

Rev Mat Complut

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Let K be an algebraically closed field and let K((X Q )) denote the field of generalized series with coefficients in K. We propose definitions of the local Łojasiewicz exponent of [191][192][193][194][195][196][197] 1997), and prove some basic properties of such numbers. Namely, we show that in both cases the exponent is attained on a parametrization of a component of F (Theorems 6 and 7), thus being a rational number. To this end, we define the notion of the Łojasiewicz pseudoexponent of F ∈ (K((X Q )) [Y ]) m for which we give a description of all the generalized series that extract the pseudoexponent, in terms of their jets. In particular, we show that there exist only finitely many jets of generalized series giving the pseudoexponent of F (Theorem 5). The main tool in the proofs is the algebraic version of Newton's Polygon Method. The results are illustrated with some explicit examples.

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Section: This Exponent Is Called the Local łOjasiewicz Exponent Of F mentioning

confidence: 99%

“…Example 3 A. Płoski in [16] proved that a rational number is equal to the Łojasiewicz exponent of a holomorphic mapping of C 2 if and only it appears in the sequence …”

Section: T)mentioning

confidence: 99%

Section: T)mentioning

confidence: 99%

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The Łojasiewicz exponent over a field of arbitrary characteristic

Brzostowski

Rodak

2015

Rev Mat Complut

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“…By the semicontinuity of the Lojasiewicz exponent in holomorphic µ-constant families of isolated singularities [20,22], we find that…”

Section: Proofs Of the Theorem 3 Corollary 4 And Corollarymentioning

confidence: 95%

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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“…This note was inspired by a recent article of Płoski [9] concerning the semi-continuity of the Łojasiewicz exponent in a family of multiplicity-constant deformation of a finite holomorphic germ. Actually, our main tool is a method of Płoski used already in [8].…”

Section: Introductionmentioning

confidence: 99%