Effective formulas for the Łojasiewicz exponent at infinity

Rodak, Tomasz; Spodzieja, Stanisław

doi:10.1016/j.jpaa.2009.01.016

Cited by 9 publications

(5 citation statements)

References 32 publications

(38 reference statements)

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“…2). A similar result for the Łojasiewicz exponent at infinity was given in [38]. The basic idea of the proof of Theorem 7 is to consider the mapping…”

Section: Introductionmentioning

confidence: 68%

Effective formulas for the local Łojasiewicz exponent

Rodak

Spodzieja

2010

Math. Z.

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We give an effective formula for the local Łojasiewicz exponent of a polynomial mapping. Moreover, we give an algorithm for computing the local dimension of an algebraic variety.Keywords Łojasiewicz exponent · Effective formulas · Germ of algebraic set · Dimension. For the properties of the mappings finite at 0 see [19].The local Łojasiewicz exponent of a mapping F : (C n , 0) → (C m , 0) is defined to be the infimum of the set of all exponents ν ∈ R in the Łojasiewicz inequality |F(z)| ≥ C|z| ν as |z| → 0 for some constant C > 0, and denoted by L 0 (F).The local Łojasiewicz exponent is an important tool in singularity theory (see [3][4][5]11,[13][14][15][16][17][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][40][41][42]). For instance, this exponent is strongly related to the order function ν I ( f ) (see [28]) and polar quotients (see [16,17,32,33,42]). The degree of C 0 -sufficiency of the jet determined by a holomorphic function f : (C n , 0) → (C, 0) is equal to [L 0 (grad f )] + 1

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“…2). A similar result for the Łojasiewicz exponent at infinity was given in [38]. The basic idea of the proof of Theorem 7 is to consider the mapping…”

Section: Introductionmentioning

confidence: 68%

Effective formulas for the local Łojasiewicz exponent

Rodak

Spodzieja

2010

Math. Z.

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“…Łojasiewicz inequalities are an important and useful tool in differential equations, singularity theory and optimization (see for instance [18] in the local case and [24] [25] at infinity). In these considerations, an estimations of the local and global Łojasiewicz exponents (see for instance [17], [20], [23], [24], [32] in the local case and [10], [12], [15], [16], [22] at infinity) play a central role. In the complex case, an essential estimations of the Łojasiewicz exponent at infinity of a polynomial mapping F = (f 1 , .…”

Section: Introductionmentioning

confidence: 99%

Metric Properties of Semialgebraic Mappings

Kurdyka

Spodzieja

Szlachcińska

2016

Discrete Comput Geom

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We give an effective estimation from above for the local Łojasiewicz exponent for separation of semialgebraic sets and for a semialgebraic mapping on a closed semialgebraic set. We also give an effective estimation from below of the Łojasiewicz exponent in the global separation for semialgebraic sets and estimation of the Łojasiewicz exponent at infinity of a semialgebraic mapping similar to the Jelonek result [14] in the complex case. Moreover, we prove that both local and global Łojasiewicz exponent of an overdetermined semialgebraic mapping F : X → R m on a closed semialgebraic set X ⊂ R n (i.e. m > dim X) are equal to the Łojasiewicz exponent of the composition L • F : X → R n for the generic linear mapping L : R m → R k , where k = dim X.

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“…On the other hand, Example 3.3 shows that it is not always as easy. But the author is still convinced that generally such recursive formulae exist (perhaps using the Łojasiewicz exponent at infinity which itself is computable, see [15]). …”

Section: Remarksmentioning

confidence: 99%

Algebra of bounded polynomials on a set Zariski closed at infinity cannot be finitely generated

Michalska

2013

Bulletin des Sciences Mathématiques

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We prove that if a set S ⊂ R n is Zariski closed at infinity, then the algebra of polynomials bounded on S cannot be finitely generated. It is a new proof of a fact already known to Plaumann and Scheiderer (2012) [1]. On the way we show that if the ring R[ζ 1 , . . . , ζ k ] ⊂ R[X] contains the ideal (ζ 1 , . . . , ζ k )R[X], then the mapping (ζ 1 , . . . , ζ k ) : R n → R k is finite.

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Effective formulas for the Łojasiewicz exponent at infinity

Abstract: a b s t r a c tWe give effective formulas for the Łojasiewicz exponent at infinity of an arbitrary complex polynomial mapping.

Cited by 9 publications

References 32 publications

Effective formulas for the local Łojasiewicz exponent

Effective formulas for the local Łojasiewicz exponent

Metric Properties of Semialgebraic Mappings

Algebra of bounded polynomials on a set Zariski closed at infinity cannot be finitely generated

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