Łojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

Bivià-Ausina, Carles; Encinas, Santiago

doi:10.1007/s13163-012-0104-0

Cited by 8 publications

(17 citation statements)

References 24 publications

(68 reference statements)

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“…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: 95%

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: T)mentioning

confidence: 95%

The Łojasiewicz exponent over a field of arbitrary characteristic

Brzostowski

Rodak

2015

Rev Mat Complut

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“…Fact II [4,Proposition 4.1] or [3,Corollary 4.7]. Let f : (C n , 0) → (C, 0) be a SQH function with principal part f 0 .…”

Section: Resultsmentioning

confidence: 99%

The Łojasiewicz exponent of semiquasihomogeneous singularities

Brzostowski

2015

Bull. London Math. Soc.

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Let f : (C n , 0) → (C, 0) be a semiquasihomogeneous function. We give a formula for the local Łojasiewicz exponent L0(f ) of f , in terms of weights of f . In particular, in the case of a quasihomogeneous (QH) isolated singularity f , we generalize a formula for L0(f ) of Krasiński, Oleksik and Płoski from 3 to n dimensions. This was previously announced in the paper [19] of Tan, Yau and Zuo [Łojasiewicz inequality for weighted homogeneous polynomial with isolated singularity, Proc. Amer. Math. Soc. 138 (2010) 3975-3984], but as a matter of fact it was not proved correctly there, as noted by the AMS reviewer Tadeusz Krasiński.As a consequence of our result, we obtain that the Łojasiewicz exponent is invariant in topologically trivial families of singularities coming from a QH germ. This is an affirmative partial answer to Teissier's conjecture.

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“…This inclusion is equivalent to saying that e(I e n−1 (I) ) = e(I e n−1 (I) + m e(I) ), by the Rees' multiplicity theorem. (see [7,27]). …”

Section: And Equality Holds If and Only Ifmentioning

confidence: 98%