Monodromies at infinity of non-tame polynomials

Takeuchi, Kiyoshi; Tibăr, Mihai

doi:10.24033/bsmf.2720

Cited by 11 publications

(11 citation statements)

References 31 publications

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“…In this section, we recall the constructions of some smooth compactifications of C n in Zaharia [30] and Takeuchi-Tibăr [26]…”

Section: Some Compactifications Of C Nmentioning

confidence: 99%

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Bifurcation values of polynomial functions and perverse sheaves

Takeuchi¹

2020

Annales De l'Institut Fourier

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We characterize bifurcation values of polynomial functions by using the theory of perverse sheaves and their vanishing cycles. In particular, by introducing a method to compute the jumps of the Euler characteristics with compact support of their fibers, we confirm the conjecture of Némethi-Zaharia in many cases.Résumé. -Nous caractérisons les valeurs de bifurcation de fonctions polynomiales en utilisant la théorie des faisceaux pervers et leurs cycles évanescents. En particulier, en introduisant une méthode pour calculer les sauts de caractéristiques d'Euler à support compact de leurs fibres, nous confirmons la conjecture de Némethi-Zaharia dans de nombreux cas.

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“…In this section, we recall the constructions of some smooth compactifications of C n in Zaharia [30] and Takeuchi-Tibăr [26]…”

Section: Some Compactifications Of C Nmentioning

confidence: 99%

“…From now we recall the smooth compactifications of C n in [26] and [30] (for their applications to monodromies at infinity see [6,17,18] and [26]). Assume that the polynomial f…”

Section: Definition 33 ([13]mentioning

confidence: 99%

Bifurcation values of polynomial functions and perverse sheaves

Takeuchi¹

2020

Annales De l'Institut Fourier

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“…We refer to [1] or [32] for generalization in dimension ≥ 3. It follows from these references and Theorem 3.16 that We recall some notions of [29, §4] (see also constructions of Matsui, Takeuchi and Tibȃr in [23], [24] and [31]…”

Section: λ-Invariantmentioning

confidence: 99%

Newton Transformations and the Motivic Milnor Fiber of a Plane Curve

Cassou-Noguès¹,

Raibaut²

2018

Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

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In this article we give an expression of the motivic Milnor fiber at infinity and the motivic nearby cycles at infinity of a polynomial f in two variables with coefficients in an algebraic closed field of characteristic zero. This expression is given in terms of some motives associated to the faces of the Newton polygons appearing in the Newton algorithm at infinity of f without any condition of convenience or non degeneracy. In the complex setting, we compute the Euler characteristic of the generic fiber of f in terms of the area of the surfaces associated to faces of the Newton polygons. Furthermore, if f has isolated singularities, we compute similarly the classical invariants at infinity λc(f ) which measures the non equisingularity at infinity of the fibers of f in P 2 , and we prove the equality between the topological and the motivic bifurcation sets and give an algorithm to compute them.

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“…In particular, in the case of the monodromy at infinity and monodromy of the cohomology of the Milnor fiber, one may obtain explicit formulas that extend algorithms given by Esterov and Takeuchi in [25]. Secondly, in the case when f is not convenient, one may use the results above to obtain formulas extending algorithms of Takeuchi and Tibar in [59]. Lastly, analogous formulas to those in Section 6.3 for families of hypersurfaces of an affine toric variety (rather than simply affine space) may be obtained using the setup and results of Steenbrink in [58].…”

Section: Introductionmentioning

confidence: 96%

Formulas for monodromy

Stapledon

2017

Res Math Sci

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Given a family X of complex varieties degenerating over a punctured disk, one is interested in computing related invariants called the motivic nearby fiber and the refined limit mixed Hodge numbers, both of which contain information about the induced action of monodromy on the cohomology of a fiber of X . Our first main result is that the motivic nearby fiber of X can be computed by first stratifying X into locally closed subvarieties that are nondegenerate in the sense of Tevelev, and then applying an explicit formula on each piece of the stratification that involves tropical geometry. Our second main result is an explicit combinatorial formula for the refined limit mixed Hodge numbers in the case when X is a family of nondegenerate hypersurfaces. As an application, given a complex polynomial, then, under appropriate conditions, we give a combinatorial formula for the Jordan block structure of the action of monodromy on the cohomology of the Milnor fiber, generalizing a famous formula of Varchenko for the associated eigenvalues. In addition, we give a formula for the Jordan block structure of the action of monodromy at infinity.

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Monodromies at infinity of non-tame polynomials

Cited by 11 publications

References 31 publications

Bifurcation values of polynomial functions and perverse sheaves

Bifurcation values of polynomial functions and perverse sheaves

Newton Transformations and the Motivic Milnor Fiber of a Plane Curve

Formulas for monodromy

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