Abstract
We consider different notions of non-degeneracy, as introduced by Kouchnirenko (NND), Wall (INND) and Beelen-Pellikaan (WNND) for plane curve singularities {f(x,y)=0} and introduce the new notion of weighted homogeneous Newton non-degeneracy (WHNND). It is known that the Milnor number μ resp. the delta-invariant δ can be computed by explicit formulas μ N resp. δ N from the Newton diagram of f if f is NND resp. WNND. It was however unknown whether the equalities μ=μ N resp. δ=δ N can be characterized by a certain non-degeneracy condition on f and, if so, by which one. We show that μ=μ N resp. δ=δ N is equivalent to INND resp. WHNND and give some applications and interesting examples related to the existence of “wild vanishing cycles”. Although the results are new in any characteristic, the main difficulties arise in positive characteristic.
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Beelen, P., Pellikaan, R.: The Newton polygon of plane curves with many rational points. Des. Codes Cryptogr. 21, 41–67 (2000)
Bivià-Ausina, C.: Local Łojasiewicz exponents, Milnor numbers and mixed multiplicities of ideals. Math. Z. 262(2), 389–409 (2009)
Boubakri, Y., Greuel, G.-M., Markwig, T.: Invariants of hypersurface singularities in positive characteristic. Rev. Mat. Complut. (2010). doi:10.1007/s13163-010-0056-1, 23 pages
Brieskorn, E., Knörrer, H.: Plane Algebraic Curves. Birkhäuser, Basel (1986), 721 pages
Campillo, A.: Algebroid Curves in Positive Characteristic. Lecture Notes in Math., vol. 613. Springer, Berlin (1980), 168 pages
Deligne, P.: La formule de Milnor. In: Sém. Géom. Algébrique du Bois-Marie, 1967–1969, SGA 7 II, Expose XVI. Lecture Notes in Math., vol. 340, pp. 197–211 (1973)
Greuel, G.-M., Lossen, C., Shustin, E.: Introduction to Singularities and Deformations. Math. Monographs. Springer, Berlin (2006), 476 pages
Greuel, G.-M., Pfister, G.: A Singular Introduction to Commutative Algebra, 2nd edn. Springer, Berlin (2008), 702 pages
Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3.1.0, A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2005). http://www.singular.uni-kl.de
Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32, 1–31 (1976)
Luengo, I.: The μ-constant stratum is not smooth. Invent. Math. 90, 139–152 (1987)
Melle-Hernández, A., Wall, C.T.C.: Pencils of curves on smooth surfaces. Proc. Lond. Math. Soc., Ser. III 83(2), 257–278 (2001)
Wall, C.T.C.: Newton polytopes and non-degeneracy. J. Reine Angew. Math. 509, 1–19 (1999)
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Greuel, GM., Nguyen, H.D. Some remarks on the planar Kouchnirenko’s theorem. Rev Mat Complut 25, 557–579 (2012). https://doi.org/10.1007/s13163-011-0082-7
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DOI: https://doi.org/10.1007/s13163-011-0082-7
Keywords
- Milnor number
- Delta-invariant
- Newton non-degenerate
- Inner Newton non-degenerate
- Weak Newton non-degenerate