Recherches sur la méthode de graeffe et les zéros des polynomes et des séries de laurent

Ostrowski, Alexander

doi:10.1007/bf02546330

Cited by 42 publications

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“…The constants in the definition of U f from Theorem 1.5 are optimal: Assertion (c) of Corollary 2.3 in Section 2 below reveals that the log 2 in the definition of α f and β f can not be replaced by any smaller constant, and Lemma 2.4 from Section 2 shows that the log 3 can not be replaced by any smaller constant. ⋄ In Section 2 we also discuss how the neighborhood U f improves (or complements) earlier root norm estimates in [Had93,Ost40a,AGS16], and how the Λ Γ provide an Archimedean version of tropical intersection multiplicity.…”

Section: Introductionmentioning

confidence: 90%

“…Special thanks go to Jan-Erik Björk and Jean-Yves Welschinger for respectively bringing Ostrowski's paper [Ost40a] and Viro's paper [Vir01] to our attention, to Matt Young for pointing out the relevance of Möbius inversion for counting lattice directions, and to Jens Forsgård for pointing out Pellet's Theorem.…”

Section: Proving Theorem 118mentioning

confidence: 99%

“…⋄ For instance, in Examples 1.2 and 1.3, ArchTrop(f ) was, respectively, a plane in R 3 equal to Amoeba(f ), and then a pair of points around which Amoeba(f ) clustered. While ArchTrop(f ) has appeared under different guises in earlier work (see, e.g., [Ost40a,Mik04,PR04,PRS11,TdW13]), explicit metric estimates for how well ArchTrop(f ) approximates Amoeba(f ) in arbitrary dimension have not yet appeared in the literature. (v − log 3, v + log 3), and let Λ Γ be the sum of λ(L) over all edges L of ArchNewt(f ) with outer normal (v, −1) satisfying v ∈ Γ.…”

Section: Introductionmentioning

confidence: 99%

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Metric estimates and membership complexity for Archimedean amoebae and tropical hypersurfaces

Avendaño

Kogan

Nisse

et al. 2018

Journal of Complexity

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Abstract. Given any complex Laurent polynomial f , Amoeba(f ) is the image of its complex zero set under the coordinate-wise log absolute value map. We give an efficiently constructible polyhedral approximation, ArchTrop(f ), of Amoeba(f ), and derive explicit upper and lower bounds, solely as a function of the number of monomial terms of f , for the Hausdorff distance between these two sets. We also show that deciding whether a given point lies in ArchTrop(f ) is doable in polynomial-time, for any fixed dimension, unlike the corresponding problem for Amoeba(f ), which is NP-hard already in one variable. ArchTrop(f ) can thus serve as a canonical low order approximation to start any higher order iterative polynomial system solving algorithm, such as homotopy continuation. ArchTrop(f ) also provides an Archimedean analogue of Kapranov's Non-Archimedean Amoeba Theorem and a higher-dimensional extension of earlier estimates of Mikhalkin and Ostrowski.In memory of Mikael Passare.

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Section: Introductionmentioning

confidence: 90%

Section: Proving Theorem 118mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Metric estimates and membership complexity for Archimedean amoebae and tropical hypersurfaces

Avendaño

Kogan

Nisse

et al. 2018

Journal of Complexity

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“…It was introduced independently by Lobachevsky, Dandelin and by Graeffe. See the papers [61], [62] by Ostrowski. Clearly, the zeros of…”

Section: Some Matrix Algebrasmentioning

confidence: 99%

“…, that is a generalization to the case of matrix polynomials of the celebrated Graeffe-Lobachevsky-Dandelin iteration [61], [62].…”

Section: Wiener-hopf Factorization and Matrix Equationsmentioning

confidence: 99%

Matrix structures and applications

Бини¹

2014

Cours du CIRM

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Perturbation of complex polynomials and normal operators

Rainer

2009

Mathematische Nachrichten

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Key words Regular roots of polynomials, absolute continuity, perturbation of normal operators MSC (2000) Primary: 26C10, 30C15, 47A55, 47A56We study the regularity of the roots of complex monic polynomials P (t) of fixed degree depending smoothly on a real parameter t. We prove that each continuous parameterization of the roots of a generic C ∞ curve P (t) (which always exists) is locally absolutely continuous. Generic means that no two of the continuously chosen roots meet of infinite order of flatness. Simple examples show that one cannot expect a better regularity than absolute continuity. This result will follow from the proposition that for any t0 there exists a positive integer N such that t → P t0 ± (t − t0) N admits smooth parameterizations of its roots near t0. We show that C n curves P (t) (where n = deg P ) admit differentiable roots if and only if the order of contact of the roots is ≥ 1. We give applications to the perturbation theory of normal matrices and unbounded normal operators with compact resolvents and common domain of definition: The eigenvalues and eigenvectors of a generic C ∞ curve of such operators can be arranged locally in an absolutely continuous way.

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Recherches sur la méthode de graeffe et les zéros des polynomes et des séries de laurent

Cited by 42 publications

References 0 publications

Metric estimates and membership complexity for Archimedean amoebae and tropical hypersurfaces

Metric estimates and membership complexity for Archimedean amoebae and tropical hypersurfaces

Matrix structures and applications

Perturbation of complex polynomials and normal operators

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