A global Łojasiewicz inequality for algebraic varieties

Ji, Shanyu; Kollár, János; Shiffman, Bernard

doi:10.1090/s0002-9947-1992-1046016-6

Cited by 48 publications

(37 citation statements)

References 7 publications

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“…[2], see also [4] and [3]). By dist(x, V ) we mean the distance from x ∈ R n to the set V ⊂ R n in a metric induced by the norm where α i,j ∈ R. An important tool in the proofs is the following result of S. Spodzieja and A. Szlachcińska (see [13,Theorem 3], cf.…”

Section: Introductionmentioning

confidence: 88%

Extensions of real regular mappings and the Łojasiewicz exponent at infinity

Osińska-Ulrych

Skalski

Spodzieja

2013

Bulletin des Sciences Mathématiques

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Let V ⊂ R n be an algebraic set of positive dimension and let L ∞ (F ) be the Łojasiewicz exponent at infinity of a regular mapping F : V → R m . We prove that F has a polynomial extension G : R n → R m such that L ∞ (G) = L ∞ (F ). Moreover, we give an estimate of the degree of this extension. Additionally, we prove that if dim V < n − 2, then for any β ∈ Q, β < L ∞ (F ) the mapping F has a polynomial extension G with L ∞ (G) = β. We also estimate its degree.

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Section: Introductionmentioning

confidence: 88%

Extensions of real regular mappings and the Łojasiewicz exponent at infinity

Osińska-Ulrych

Skalski

Spodzieja

2013

Bulletin des Sciences Mathématiques

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“…Proof According to Łojasiewicz Inequality as in [37], the following property holds for every non-constant homogeneous polynomial f ∈ C[X 0 , . .…”

Section: Lemma 21 (Complex Differentiation Under the Integral Sign)mentioning

confidence: 99%

On the zeta Mahler measure function of the Jacobian determinant, condition numbers and the height of the generic discriminant

Pardo

2016

AAECC

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authors introduced zeta Mahler measure functions for multivariate polynomials [Cassaigne and Maillot (J Number Theory 83:226-255, 2000) called them "zeta Igusa" functions, but we follow here the terminology of Akatsuka (J Number Theory 129:2713-2734]. We generalize this notion by defining a zeta Mahler measure function Z X (·, f ) : C −→ C, where X is a compact probability space and f : X −→ C is a function bounded almost everywhere in X . We give sufficient conditions that imply that this function is holomorphic in certain domains. Zeta Mahler measure functions contains big amounts of information about the expected behavior of f on X . This generalization is motivated by the study of several quantities related to numerical methods that solve systems of multivariate polynomial equations. We study the functions Z(·, 1 / · aff ), Z(·, 1 /μ norm ) and Z(·, JAC), respectively associated to the norm of the affine zeros ( · aff ), the non-linear condition number (μ norm ) and the Jacobian determinant (JAC) of complete intersection zero-dimensional projective varieties. We find the exact value of these functions in terms of Gamma functions and we also describe their respective domains of holomorphy in C. With the exact value of these zeta functions we can immediately prove and exhibit expectations of some average properties of zero-dimensional algebraic varieties. For instance, the exact knowledge of Z(·, 1 / · aff ) yields as a consequence that the expectation of the mean of the logarithm of the norms of the affine zeros of a random system of polynomial equations is one half of the nth harmonic number H n . Other conclusions are exhibited along the manuscript. Using these generalized zeta functions we exhibit the exact value of the arithmetic height of the hyper-surface known as the discriminant variety (roughly speaking the variety formed by all systems of equations having a singular zero).

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“…The answer to these questions is "yes" provided the matrices are drawn from a compact set (as we already noted in [13,Remark 1]). The main new idea proposed here is to gain a better understanding of this issue by using the Łojasiewicz inequality [16]. Given a function (usually a polynomial or a maximum of polynomials) which takes a small value at a given point, the Łojasiewicz inequality provides an upper bound on the distance from this point to the set of zeros of the function.…”

Section: Introductionmentioning

confidence: 99%

Robust stability conditions for switched linear systems: Commutator bounds and the Łojasiewicz inequality

Baryshnikov

Liberzon

2013

52nd IEEE Conference on Decision and Control

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This paper discusses conditions for stability of switched linear systems under arbitrary switching, formulated in terms of smallness of appropriate commutators of the matrices generating the switched system. Such conditions provide robust variants of well-known stability conditions requiring these commutators to vanish and leading to the existence of a common quadratic Lyapunov function. The main contribution of the paper is to apply the Łojasiewicz inequality to characterize the persistence of a common quadratic Lyapunov function as the matrices are perturbed so that their commutators no longer vanish but instead are sufficiently small. It is shown how known constructions of common quadratic Lyapunov functions for commuting matrices and for matrices generating nilpotent or solvable Lie algebras can be used, in conjunction with the Łojasiewicz inequality, to estimate allowable deviations of the commutators from zero.

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A global Łojasiewicz inequality for algebraic varieties

Cited by 48 publications

References 7 publications

Extensions of real regular mappings and the Łojasiewicz exponent at infinity

Extensions of real regular mappings and the Łojasiewicz exponent at infinity

On the zeta Mahler measure function of the Jacobian determinant, condition numbers and the height of the generic discriminant

Robust stability conditions for switched linear systems: Commutator bounds and the Łojasiewicz inequality

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A global Łojasiewicz inequality for algebraic varieties

Cited by 48 publications

References 7 publications

Extensions of real regular mappings and the Łojasiewicz exponent at infinity

Extensions of real regular mappings and the Łojasiewicz exponent at infinity

On the zeta Mahler measure function of the Jacobian determinant, condition numbers and the height of the generic discriminant

Robust stability conditions for switched linear systems: Commutator bounds and the &#x0141;ojasiewicz inequality

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Robust stability conditions for switched linear systems: Commutator bounds and the Łojasiewicz inequality