Singular Points of Complex Hypersurfaces. (AM-61)

Milnor, John

doi:10.1515/9781400881819

Cited by 908 publications

(953 citation statements)

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“…Remember that the zeta function of monodromy associated to an isolated singularity in odd dimensions is a polynomial; in even dimensions it is (1 − t) divided by a polynomial. Indeed, for isolated singularities, Milnor proved that the cohomology groups H i (F, C) are {0} for i = 0 and i = n − 1, where n denotes the dimension (see [Mi68]). …”

Section: Theorem 2 ([Var76]) the Zeta Function Of Monodromy Of F In mentioning

confidence: 99%

Monodromy conjecture for nondegenerate surface singularities

Lemahieu¹,

Proeyen

2011

Trans. Amer. Math. Soc.

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Abstract. We prove the monodromy conjecture for the topological zeta function for all nondegenerate surface singularities. Fundamental in our work is a detailed study of the formula for the zeta function of monodromy by Varchenko and the study of the candidate poles of the topological zeta function yielded by what we call 'B 1 -facets'. In particular, new cases among the nondegenerate surface singularities for which the monodromy conjecture is now proven are the nonisolated singularities, the singularities giving rise to a topological zeta function with multiple candidate poles and the ones for which the Newton polyhedron contains a B 1 -facet.

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Section: Theorem 2 ([Var76]) the Zeta Function Of Monodromy Of F In mentioning

confidence: 99%

Monodromy conjecture for nondegenerate surface singularities

Lemahieu¹,

Proeyen

2011

Trans. Amer. Math. Soc.

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“…Recall also that in the above situation, as in the case of polynomial functions over C n (see Milnor [31]), we can define the Milnor monodromy operators…”

Section: Proposition 23 ([8 Proposition 422]) There Exists a Natmentioning

confidence: 99%

“…For the classical case where X = C m see [1,3,21,22,29,31,34,45,47] etc. Similarly, also for any y ∈ X 0 = {x ∈ X | f(x) = 0} we can define the Milnor fiber F y and its monodromies Φ j,y .…”

Section: Proposition 23 ([8 Proposition 422]) There Exists a Natmentioning

confidence: 99%

Motivic Milnor Fibers over Complete Intersection Varieties and their Virtual Betti Numbers

Esterov

Takeuchi

2011

International Mathematics Research Notices

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“…Sets expressed in this way are called semialgebraic and are known to have a stratification with special properties [2,27]. One can show, for example, that the sets A mj (j even) and cl A m are semialgebraic.…”

Section: Smoothness Assumptions Onmentioning

confidence: 99%

Convex duality and entropy-based moment closures: Characterizing degenerate densities

Hauck

Levermore

Tits

2008

2008 47th IEEE Conference on Decision and Control

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Abstract.A common method for constructing a function from a finite set of moments is to solve a constrained minimization problem. The idea is to find, among all functions with the given moments, that function which minimizes a physically motivated, strictly convex functional. In the kinetic theory of gases, this functional is the kinetic entropy; the given moments are macroscopic densities; and the solution to the constrained minimization problem is used to formally derive a closed system of partial differential equations which describe how the macroscopic densities evolve in time. Moment equations are useful because they simplify the kinetic, phase-space description of a gas, and with entropy-based closures, they retain many of the fundamental properties of kinetic transport. Unfortunately, in many situations, macroscopic densities can take on values for which the constrained minimization problem does not have a solution. Essentially, this is because the moments are not continuous functionals with respect to the L 1 topology. In this paper, we give a geometric description of these so-called degenerate densities in the most general possible setting. Our key tool is the complementary slackness condition that is derived from a dual formulation of a minimization problem with relaxed constraints. We show that the set of degenerate densities is a union of convex cones and, under reasonable assumptions, that this set is small in both a topological and a measuretheoretic sense. This result is important for further assessment and implementation of entropy-based moment closures.

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Singular Points of Complex Hypersurfaces. (AM-61)

Cited by 908 publications

References 0 publications

Monodromy conjecture for nondegenerate surface singularities

Monodromy conjecture for nondegenerate surface singularities

Motivic Milnor Fibers over Complete Intersection Varieties and their Virtual Betti Numbers

Convex duality and entropy-based moment closures: Characterizing degenerate densities

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