Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials

D'Acunto, Didier; Kurdyka, Krzysztof

doi:10.4064/ap87-0-5

Cited by 51 publications

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“…We also considered the following situation : See figures ( 5) and ( 6) and ( 7) and (8). (5.2) 5.1.3.…”

Section: Examplementioning

confidence: 99%

Asymptotics for some discretizations of dynamical systems, application to second order systems with non-local nonlinearities

Horsin¹,

Jendoubi²

2022

CPAA

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<p style='text-indent:20px;'>In the present paper we study the asymptotic behavior of discretized finite dimensional dynamical systems. We prove that under some discrete angle condition and under a Lojasiewicz's inequality condition, the solutions to an implicit scheme converge to equilibrium points. We also present some numerical simulations suggesting that our results may be extended under weaker assumptions or to infinite dimensional dynamical systems.</p>

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“…We also considered the following situation : See figures ( 5) and ( 6) and ( 7) and (8). (5.2) 5.1.3.…”

Section: Examplementioning

confidence: 99%

Asymptotics for some discretizations of dynamical systems, application to second order systems with non-local nonlinearities

Horsin¹,

Jendoubi²

2022

CPAA

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“…-Voici une version quantitative de l'inégalité de Łoja-siewicz (1.1) établie dans [11] et [10] : supposons que f soit un polynôme de degré d à n variables, alors …”

Section: Remarque 22 -Dans Le Cas Oùunclassified

Gradient horizontal de fonctions polynomiales

Đinh¹,

Kurdyka²,

Orro³

2009

Ann. inst. Fourier

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“…Then [35,36,40]). Interesting estimates of L 0 (F) are known (see [11,12,21,29,36,41]). In this connection, formulas for L 0 (F) are of interest.…”

Section: Introductionmentioning

confidence: 99%

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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Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials

Cited by 51 publications

References 0 publications

Asymptotics for some discretizations of dynamical systems, application to second order systems with non-local nonlinearities

Asymptotics for some discretizations of dynamical systems, application to second order systems with non-local nonlinearities

Gradient horizontal de fonctions polynomiales

Effective formulas for the local Łojasiewicz exponent

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