Abstract
In this article we give an expression of the motivic Milnor fiber at the origin of a polynomial in two variables with coefficients in an algebraically closed field. The expression is given in terms of some motives associated to the faces of the Newton polygons appearing in the Newton algorithm. In the complex setting, we deduce a computation of the Euler characteristic of the Milnor fiber in terms of the area of the surfaces under the Newton polygons encountered in the Newton algorithm which generalizes the Milnor number computation by Kouchnirenko in the isolated case.
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Acknowledgements
The first author is partially supported by the projects MTM2016-76868-C2-1-P (UCM Madrid) and MTM2016-76868-C2-2-P (Zaragoza). The second author is partially supported by the project ANR-15-CE40-0008 (Dfigo). The authors thank the referees for theirs suggestions.
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Cassou-Noguès, P., Raibaut, M. (2018). Newton Transformations and the Motivic Milnor Fiber of a Plane Curve. In: Greuel, GM., Narváez Macarro, L., Xambó-Descamps, S. (eds) Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics. Springer, Cham. https://doi.org/10.1007/978-3-319-96827-8_7
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