Journal of the Mathematical Society of Japan
Online ISSN : 1881-1167
Print ISSN : 0025-5645
ISSN-L : 0025-5645
Arnold's problem on monotonicity of the Newton number for surface singularities
Szymon BrzostowskiTadeusz KrasińskiJustyna Walewska
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2019 年 71 巻 4 号 p. 1257-1268

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According to the Kouchnirenko Theorem, for a generic (meaning non-degenerate in the Kouchnirenko sense) isolated singularity 𝑓 its Milnor number 𝜇 (𝑓) is equal to the Newton number 𝜈 (𝚪+(𝑓)) of a combinatorial object associated to 𝑓, the Newton polyhedron 𝚪+ (𝑓). We give a simple condition characterizing, in terms of 𝚪+ (𝑓) and 𝚪+ (𝑔), the equality 𝜈 (𝚪+(𝑓)) = 𝜈 (𝚪+(𝑔)), for any surface singularities 𝑓 and 𝑔 satisfying 𝚪+ (𝑓) ⊂ 𝚪+ (𝑔). This is a complete solution to an Arnold problem (No. 1982-16 in his list of problems) in this case.

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