The Milnor Number of Plane Branches with Tame Semigroups of Values

2018 ◽  
Vol 49 (4) ◽  
pp. 789-809
Author(s):  
A. Hefez ◽  
J. H. O. Rodrigues ◽  
R. Salomão
Keyword(s):  
Milnor Number

scholarly journals Conjectures on Stably Newton Degenerate Singularities

2021 ◽  
Author(s):  
Jan Stevens
Keyword(s):  
Characteristic Zero ◽  
Arbitrary Number ◽  
Milnor Number ◽  
Plane Curves ◽  
Newton Diagram ◽  
Finite Characteristic ◽  
Vanishing Cycles

AbstractWe 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.

scholarly journals The Image Milnor Number and Excellent Unfoldings

2021 ◽  
Author(s):  
R Giménez Conejero ◽  
J J Nuño-Ballesteros
Keyword(s):  
Milnor Number ◽  
Basic Properties ◽  
Topological Invariance

Abstract We show three basic properties of the image Milnor number µI(f) of a germ $f\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with isolated instability. First, we show the conservation of the image Milnor number, from which one can deduce the upper semi-continuity and the topological invariance for families. Second, we prove the weak Mond’s conjecture, which states that µI(f) = 0 if and only if f is stable. Finally, we show a conjecture by Houston that any family $f_t\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with $\mu_I(\,f_t)$ constant is excellent in Gaffney’s sense. For technical reasons, in the last two properties, we consider only the corank 1 case.

scholarly journals The jump of the Milnor number in the X 9 singularity class

2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Szymon Brzostowski ◽  
Tadeusz Krasiński
Keyword(s):  
Milnor Number ◽  
Isolated Singularity ◽  
Milnor Numbers

AbstractThe 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 deformations (f s). We prove that for the singularities in the X 9 singularity class their jumps are equal to 2.

Singularity of Monomial Curves in A3 and Gorenstein Monomial Curves in A4

1985 ◽  
Vol 37 (5) ◽  
pp. 872-892 ◽  
Author(s):  
Jürgen Kraft
Keyword(s):  
Correction Term ◽  
Milnor Number ◽  
Monomial Curves ◽  
Space Curves ◽  
Image Position ◽  
Monomial Curve

Let 2 ≦ s ∊ N and {n1, …, ns) ⊆ N*. In 1884, J. Sylvester [13] published the following well-known result on the singularity degree S of the monomial curve whose corresponding semigroup is S: = 〈n1, …, ns): If s = 2, thenLet K: = –Z\S andfor all 1 ≦ i ≦ s. We introduce the invariantof S involving a correction term to the Milnor number 2δ [4] of S. As a modified version and extension of Sylvester's result to all monomial space curves, we prove the following theorem: If s = 3, thenWe prove similar formulas for s = 4 if S is symmetric.

Double points in families of map germs from ℝ2 to ℝ3

2020 ◽  
pp. 1-18
Author(s):  
J. A. Moya-Pérez ◽  
J. J. Nuño-Ballesteros
Keyword(s):  
Double Point ◽  
Parameter Family ◽  
Milnor Number ◽  
Double Points ◽  
The Family ◽  
Number Formula ◽  
Real Analytic

We show that a 1-parameter family of real analytic map germs [Formula: see text] with isolated instability is topologically trivial if it is excellent and the family of double point curves [Formula: see text] in [Formula: see text] is topologically trivial. In particular, we deduce that [Formula: see text] is topologically trivial when the Milnor number [Formula: see text] is constant.

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