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

Brzostowski, Szymon; Krasiński, Tadeusz; Walewska, Justyna

doi:10.2969/jmsj/78557855

Cited by 5 publications

(4 citation statements)

References 11 publications

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“…This allows us to construct the desired simultaneous resolution. In this section, we give an affirmative answer to the conjecture presented in [BKW19]. This result together with Theorem 2.4 (see [Fur04]) is a complete solution to an Arnold problem (No.…”

Section: Introductionsupporting

confidence: 61%

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Newton non-degenerate -constant deformations admit simultaneous embedded resolutions

Leyton-Álvarez¹,

Mourtada²,

Spivakovsky³

2022

Compositio Math.

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Let $ {\mathbb {C}}^{n+1}_o$ denote the germ of $ {\mathbb {C}}^{n+1}$ at the origin. Let $V$ be a hypersurface germ in $ {\mathbb {C}}^{n+1}_o$ and $W$ a deformation of $V$ over $ {\mathbb {C}}_{o}^{m}$ . Under the hypothesis that $W$ is a Newton non-degenerate deformation, in this article we prove that $W$ is a $\mu$ -constant deformation if and only if $W$ admits a simultaneous embedded resolution. This result gives a lot of information about $W$ , for example, the topological triviality of the family $W$ and the fact that the natural morphism $(\operatorname {W( {\mathbb {C}}_{o})}_{m})_{{\rm red}}\rightarrow {\mathbb {C}}_{o}$ is flat, where $\operatorname {W( {\mathbb {C}}_{o})}_{m}$ is the relative space of $m$ -jets. On the way to the proof of our main result, we give a complete answer to a question of Arnold on the monotonicity of Newton numbers in the case of convenient Newton polyhedra.

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Section: Introductionsupporting

confidence: 61%

“…The following Theorem generalizes to all dimensions the main theorem of [BKW19]. In [BKW19] this result is conjectured. Definition 2.14.…”

Section: By Construction We Obtainmentioning

confidence: 71%

Newton non-degenerate -constant deformations admit simultaneous embedded resolutions

Leyton-Álvarez¹,

Mourtada²,

Spivakovsky³

2022

Compositio Math.

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“…He defines an exceptional face Δ of Γ ( f ) ⊂ R n as a facet with one of its vertices at a distance 1 to a coordinate axis, while the remaining vertices define an (n −2)-dimensional face in one of the coordinate hyperplanes through that axis. A combinatorial characterisation of Newton polyhedra Γ + ⊂ Γ + in R 3 + with ν(Γ − ) = ν(Γ − ) Brzostowski, Krasiński and Walewska [8].…”

Section: Definition 18mentioning

confidence: 99%

Conjectures on Stably Newton Degenerate Singularities

Stevens

2021

Arnold Math J.

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We discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

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“…The proof of our result is based on [3], where we gave an explicit formula for the Łojasiewicz exponent of a non-degenerate surface singularity in terms of its Newton polyhedron (see formula (2.1) below), and on [4] which gives a characterization of µ-constant non-degenerate families of surface singularities. We notice that the characterization from [4] has recently been extended (in an equivalent form) to any dimension by M. Leyton-Álvarez, H. Mourtada and M. Spivakovsky [13, Thm. 2.15].…”

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confidence: 99%