Puiseux parametric equations of analytic sets

Aroca, Fuensanta

doi:10.1090/s0002-9939-04-07337-x

Cited by 12 publications

(6 citation statements)

References 17 publications

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“…Corollary 2. (Aroca, [2]) Let X be an algebraic variety of C N +M , 0 ∈ X, dim(X) = N. Let U be a neighborhood of 0, and let π be the restriction to X ∩ U of the projection (z 1 , ..., z N +M ) → (z 1 , ..., z N ). Assume π is a finite morphism.…”

Section: -The Series Development Of the Parameterizationsmentioning

confidence: 99%

Puiseux Parametric Equations via the Amoeba of the Discriminant

Aroca

Saavedra

2017

Trends in Mathematics

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Given an algebraic variety we get Puiseux type parametrizations on suitable Reinhardt domains. These domains are defined using the amoeba of hypersurfaces containing the discriminant locus of a finite projection of the variety. 1.-IntroductionThe theory of complex algebraic or analytic singularities is the study of systems of a finite number of equations in the neighborhood of a point where the rank of the Jacobian matrix is not maximal. These points are called singular points.Isaac Newton in a letter to Henry Oldenburg [12], described an algorithm to compute term by term local parameterizations at singular points of plane curves. The existence of such parameterizations (i.e. the fact that the algorithm really works) was proved by Puiseux [13] two centuries later. This is known as the Newton-Puiseux theorem and asserts that we can find local parametric equations of the form z 1 = t k , z 2 = ϕ(t) where ϕ is a convergent power series.Singularities of dimension greater than one are not necessarily parameterizable. In this paper we prove that, for every connected component of the complement of the amoeba of the discriminant locus of a projection of an algebraic variety, there exist local Puiseux parametric equations of the variety. The series appearing in those parametric equations have support contained in cones which can be described in terms of the connected components of the complement of the amoeba. These cones are not necessarily strongly convex.The 2.-Polyhedral convex conesA set σ ⊆ R N is said to be a convex rational polyhedral cone when it can be expressed in the form σ = {λ 1 u (1) + λ 2 u (2) + · · · + λ M u (M ) | λ j ∈ R ≥0 },

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Section: -The Series Development Of the Parameterizationsmentioning

confidence: 99%

Puiseux Parametric Equations via the Amoeba of the Discriminant

Aroca

Saavedra

2017

Trends in Mathematics

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“…The following is an easy but frequently useful lemma (see [7,Lemma 4.7] or [52,6.VI] or [41,Lemma 4.3] for a proof) concerning the ordering of fractional normal crossings and their exponents.…”

Section: Remarksmentioning

confidence: 99%

A multi-dimensional resolution of singularities with applications to analysis

Collins¹,

Greenleaf²,

Pramanik³

2013

ajm

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Abstract. We formulate a resolution of singularities algorithm for analyzing the zero sets of real-analytic functions in dimensions ≥ 3. Rather than using the celebrated result of Hironaka, the algorithm is modeled on a more explicit and elementary approach used in the contemporary algebraic geometry literature. As an application, we define a new notion of the height of real-analytic functions, compute the critical integrability index, and obtain sharp growth rate of sublevel sets. This also leads to a characterization of the oscillation index of scalar oscillatory integrals with real-analytic phases in all dimensions.

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“…Let σ be a strongly convex cone that contains the first orthant. In [3] it is shown that (when it is not empty) the domain of convergence of a series with exponents in a strongly convex cone σ contains an open set W that has the origin as accumulation point. Moreover, by the results of [3,Prop 3.4], the intersection of a finite number of such domains is non-empty.…”

Section: The Local Parametrizations Defined By the Seriesmentioning

confidence: 99%

“…Quasi-ordinary singularities admit analytic local parametrizations. This was shown for hypersurfaces by S. Abhyankar [1] and extended to arbitrary codimension in [3]. Quasi-ordinary singularities have been the object of study of many research papers [8, 15, 32, 28, ...].…”

Section: The Solutionsmentioning

confidence: 99%

“…In the complex case, a series of positive order, obtained by the Newton-Puiseux method for algebraic plane curves, represents a local parametrization of the curve around the origin. In Section 3 we explain how an M -tuple of series arising as a solution to the general Newton-Puiseux statement also represents a local parametrization recalling results form [3].…”

Section: Introductionmentioning

confidence: 99%

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Puiseux Power Series Solutions for Systems of Equations

2010

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We give an algorithm to compute term by term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method for plane algebraic curves replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities.

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Puiseux parametric equations of analytic sets

Cited by 12 publications

References 17 publications

Puiseux Parametric Equations via the Amoeba of the Discriminant

Puiseux Parametric Equations via the Amoeba of the Discriminant

A multi-dimensional resolution of singularities with applications to analysis

Puiseux Power Series Solutions for Systems of Equations

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