scholarly journals Monodromy of a class of analytic generalized nilpotent systems through their Newton diagram

2015 ◽  
Vol 287 ◽  
pp. 78-87 ◽  
Author(s):  
A. Algaba ◽  
C. García ◽  
M. Reyes
Keyword(s):  
Newton Diagram

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.

The asymptotics of the return map of a singular point with fixed Newton diagram

1992 ◽  
Vol 60 (6) ◽  
pp. 1765-1781 ◽  
Author(s):  
F. S. Berezovskaya ◽  
N. B. Medvedeva
Keyword(s):  
Singular Point ◽  
Newton Diagram ◽  
Return Map

scholarly journals Nearly irreducibility of polynomials and the Newton diagrams

2020 ◽  
Vol 19 (1) ◽  
pp. 65-77
Author(s):  
Mateusz Masternak
Keyword(s):  
Complex Variables ◽  
Common Zero ◽  
Newton Diagram

AbstractLet f be a polynomial in two complex variables. We say that f is nearly irreducible if any two nonconstant polynomial factors of f have a common zero in C2. In the paper we give a criterion of nearly irreducibility for a given polynomial f in terms of its Newton diagram.

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