Abstract
The jump of the Milnor number of an isolated singularity \(f_{0}\) is the minimal non-zero difference between the Milnor numbers of \(f_{0}\) and one of its deformation \((f_{s}).\) In the case \(f_{s}\) are non-degenerate singularities we call the jump non-degenerate. We give a formula (an inductive algorithm using diophantine equations) for the non-degenerate jump of \(f_{0}\) in the case \(f_{0}\) is a convenient singularity with only one \((n-1)\)-dimensional face of its Newton diagram which equivalently (in our problem) can be replaced by the Brieskorn–Pham singularities.
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Arnold, V.I., Guseĭn-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, vol. I, volume 82 of Monographs in Mathematics. Birkhäuser Boston Inc., Boston, MA. The Classification of Critical Points, Caustics and Wave Fronts, Translated from the Russian by Ian Porteous and Mark Reynolds (1985)
Biviá-Ausina, C.: Local Łojasiewicz exponents, Milnor numbers and mixed multiplicities of ideals. Math. Z. 262(2), 389–409 (2009)
Bodin, A.: Jump of Milnor numbers. Bull. Braz. Math. Soc. (N.S.) 38(3), 389–396 (2007)
Brzostowski, S., Krasiński, T.: The jump of the Milnor number in the \({X}_9\) singularity class. Cent. Eur. J. Math. 12, 429–435 (2014)
Brzostowski, S., Krasiński, T., Walewska, J.: Milnor numbers in deformations of homogeneous singularities. Bull. Sci. Math. arXiv:1404.7704 (to appear)
Brzostowski, S., Oleksik, G.: On combinatorial criteria for non-degenerate singularities. Kodai Math. J. 39(2), 455–468 (2016)
Furuya, M.: Lower bound of Newton number. Tokyo J. Math. 27, 177–186 (2004)
Greuel, G.-M., Lossen, C., Shustin, E.: Introduction to Singularities and Deformations. Springer Monographs in Mathematics. Springer, Berlin (2007)
Guseĭn-Zade, S.M.: On singularities from which an \(A_1\) can be split off. Funktcional. Anal. i Prilozhen. 27(1), 68–71 (1993)
Gwoździewicz, J.: Note on the Newton number. Univ. Iagel. Acta Math. 46, 31–33 (2008)
Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32(1), 1–31 (1976)
Mondal, P.: Intersection multiplicity, Milnor number and Bernstein’s theorem. ArXiv e-prints, December 2016
Oka, M.: On the bifurcation of the multiplicity and topology of the Newton boundary. J. Math. Soc. Jpn. 31(3), 435–450 (1979)
Płoski, A.: Newton polygons and the Łojasiewicz exponent of a holomorphic mapping of \({ C}^2\). Ann. Polon. Math. 51, 275–281 (1990)
Płoski, A.: Milnor number of a plane curve and Newton polygons. Univ. Lagel. Acta Math. 37, 75–80 (1999)
Walewska, J.: The second jump of Milnor numbers. Demonstr. Math. 43(2), 361–374 (2010)
Walewska, J.: Jumps of Milnor numbers in families of non-degenerate and non-convenient singularities. In: Analytic and Algebraic Geometry, Faculty of Mathematics and Computer Science. Łódź University Press, pp. 141–153 (2013)
Acknowledgements
The authors are grateful to Andrzej Nowicki for the proof of the formula in Remark 5. We also thank Szymon Brzostowski for the discussion. The first author was partially supported by the Polish National Science Centre (NCN), Grant No. 2012/07/B/ST1/03293.
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Funding was provided by Ministerstwo Nauki i Szkolnictwa Wyższego.
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Krasiński, T., Walewska, J. Jumps of Milnor Numbers of Brieskorn–Pham Singularities in Non-degenerate Families. Results Math 73, 94 (2018). https://doi.org/10.1007/s00025-018-0849-y
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DOI: https://doi.org/10.1007/s00025-018-0849-y