Fibre de Milnor motivique à l'infini

Raibaut, Michel

doi:10.1016/j.crma.2010.01.008

Cited by 11 publications

(16 citation statements)

References 11 publications

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“…Non degenerate and convenient case). -In[27, Théorème 4.8], the second author proved that for any convenient and non-degenerate polynomial f at infinity, the motivic nearby cycle S ∞ f,c is zero for any value c in k. We can also deduce this result in dimension 2 from Theorem 3.23. Let c be in k and consider formula (3.14) of Theorem 3.23.-As f is convenient, the Newton polygon at infinity N ∞ (f ) has two zero dimensional faces (a, 0) and (0, b).…”

mentioning

confidence: 87%

“…Let k be an algebraic closed field of characteristic zero. Let f be a polynomial with coefficients in k. Using the motivic integration theory, introduced by Kontsevich in [20], and more precisely constructions of Denef -Loeser in [11,12,14] and Guibert -Loeser -Merle in [18], Matsui-Takeuchi in [23,24] and independently the second author in [29] (see also [27] and [28]) defined a motivic Milnor fiber at infinity of f , denoted by S f,∞ . It is an element of M Gm {∞}×Gm , a modified Grothendieck ring of varieties over k endowed with an action of the multiplicative group G m of k. It follows from Denef-Loeser results that the motive S f,∞ is a "motivic" incarnation of the topological Milnor fiber at infinity of f , denoted by F ∞ and endowed with its monodromy action T ∞ .…”

Section: Introductionmentioning

confidence: 99%

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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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mentioning

confidence: 87%

Section: Introductionmentioning

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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“…The motivic nearby fiber at infinity of a convenient polynomial f was introduced independently and from different perspectives in [40] and [46], and an explicit formula was given vectors of M R = R n . Then P is the convex hull of the origin and {m i f i | 1 ≤ i ≤ n}.…”

Section: Applications To the Monodromy Of Complex Polynomialsmentioning

confidence: 99%

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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“…On pourra se référer à la note [18] et au théorème [20, théorème 2.4]. Pour d'autres propriétés de cet objet on pourra se référer aux travaux en cours de Kiyoshi Takeuchi et Yutaka Matsui [16] et [15].…”

Section: Espace D'arcs Soit δ Et N Deux Entiers Strictement Positifsunclassified

“…Dans [18], [20] et [19], grâce à l'intégration motivique, nous définissons S f,∞ , la fibre de Milnor motivique de f à l'infini. Cet invariant de f , est construit à l'aide d'une compactification mais n'en dépend pas.…”

Section: Introductionunclassified

Fibre de Milnor motivique à l'infini

Cited by 11 publications

References 11 publications

Newton Transformations and the Motivic Milnor Fiber of a Plane Curve

Newton Transformations and the Motivic Milnor Fiber of a Plane Curve

Formulas for monodromy

Fibre de Milnor motivique à l’infini et composition avec un polynôme non dégénéré

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