Asymptotics and Newton polyhedra

This chapter presents three sets of problems and explains how these questions can be answered in an (almost) complete way in terms of Newton polyhedra associated to the given surface S (here, a smooth, finite type hypersurface in R³ with Riemannian surface measure dσ‎). The first problem is a classical question about estimates for oscillatory integrals, and there exists a huge body of results on it, in particular for convex hypersurfaces. The other two problems had first been formulated by Stein: the study of maximal averages along hypersurfaces has been initiated in Stein's work on the spherical maximal function, and also the idea of Fourier restriction goes back to him.

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The Method of Newton Polyhedra for Investigating Singular Positions of Some Mechanisms

Computer Algebra in Scientific Computing CASC 2001 ◽

10.1007/978-3-642-56666-0_37 ◽

2001 ◽

pp. 491-498

Author(s):

Akhmadjon Soleev ◽

Adizjon S. Barotov

Keyword(s):

Newton Polyhedra

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THE INTEGRAL CLOSURE OF MODULES, BUCHSBAUM–RIM MULTIPLICITIES AND NEWTON POLYHEDRA

Journal of the London Mathematical Society ◽

10.1112/s0024610703005155 ◽

2004 ◽

Vol 69 (02) ◽

pp. 407-427 ◽

Cited By ~ 10

Author(s):

CARLES BIVIÀ-AUSINA

Keyword(s):

Integral Closure ◽

Newton Polyhedra

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Positive Quadratic Differential Forms: Topological Equivalence Through Newton Polyhedra

Journal of Dynamical and Control Systems ◽

10.1007/s10883-006-0003-1 ◽

2006 ◽

Vol 12 (4) ◽

pp. 489-516 ◽

Cited By ~ 1

Author(s):

C. Gutierrez ◽

R. D. S. Oliveira ◽

M. A. Teixeira

Keyword(s):

Quadratic Differential ◽

Differential Forms ◽

Topological Equivalence ◽

Newton Polyhedra ◽

Quadratic Differential Forms

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Newton polyhedra (resolution of singularities)

Journal of Soviet Mathematics ◽

10.1007/bf01084822 ◽

1984 ◽

Vol 27 (3) ◽

pp. 2811-2830 ◽

Cited By ~ 9

Author(s):

A. G. Khovanskii

Keyword(s):

Resolution Of Singularities ◽

Newton Polyhedra

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Newton filtrations, graded algebras and codimension of non-degenerate ideals

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004102005911 ◽

2002 ◽

Vol 133 (1) ◽

pp. 55-75 ◽

Cited By ~ 12

Author(s):

CARLES BIVIÀ-AUSINA ◽

TOSHIZUMI FUKUI ◽

MARCELO JOSÉ SAIA

Keyword(s):

Graded Algebras ◽

Newton Polyhedra ◽

Degeneracy Condition

We investigate a generalization of the method introduced by Kouchnirenko to compute the codimension (colength) of an ideal under a certain non-degeneracy condition on a given system of generators of I. We also discuss Newton non-degenerate ideals and give characterizations using the notion of reductions and Newton polyhedra of ideals.

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