Journal of the Mathematical Society of Japan

Arnold's problem on monotonicity of the Newton number for surface singularities


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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 $\mu (f)$ is equal to the Newton number $\nu (\mathbf{\Gamma}_{+}(f))$ of a combinatorial object associated to $f$, the Newton polyhedron $\mathbf{\Gamma}_+ (f)$. We give a simple condition characterizing, in terms of $\mathbf{\Gamma}_+ (f)$ and $\mathbf{\Gamma}_+ (g)$, the equality $\nu (\mathbf{\Gamma}_{+}(f)) = \nu (\mathbf{\Gamma}_{+}(g))$, for any surface singularities $f$ and $g$ satisfying $\mathbf{\Gamma}_+ (f) \subset \mathbf{\Gamma}_+ (g)$. This is a complete solution to an Arnold problem (No. 1982-16 in his list of problems) in this case.

Article information

J. Math. Soc. Japan, Volume 71, Number 4 (2019), 1257-1268.

Received: 3 August 2017
Revised: 24 July 2018
First available in Project Euclid: 14 June 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32S25: Surface and hypersurface singularities [See also 14J17]
Secondary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 14J17: Singularities [See also 14B05, 14E15]

Arnold's problem Milnor number non-degenerate singularity Newton polyhedron


BRZOSTOWSKI, Szymon; KRASIŃSKI, Tadeusz; WALEWSKA, Justyna. Arnold's problem on monotonicity of the Newton number for surface singularities. J. Math. Soc. Japan 71 (2019), no. 4, 1257--1268. doi:10.2969/jmsj/78557855.

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