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Language: search.filters.language.EnglishSubject: search.filters.subject.Newton polygon

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    Topological type of discriminants of some special families
    (Springer Science+Business Media, 2021-09-08)
    We will describe the topological type of the discriminant curve of the morphism $$(\ell , f)$$ ( ℓ , f ) , where $$\ell $$ ℓ is a smooth curve and f is an irreducible curve (branch) of multiplicity less than five or a branch such that the difference between its Milnor number and Tjurina number is less than 3. We prove that for a branch of these families, the topological type of the discriminant curve is determined by the semigroup, the Zariski invariant and at most two other analytical invariants of the branch.
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    Characterization of second type plane foliations using Newton polygons
    (Sciendo, 2022-05-01)
    In this article we characterize the foliations that have the same Newton polygon that their union of formal separatrices, they are the foliations called of the second type. In the case of cuspidal foliations studied by Loray [ Lo ], we precise this characterization using the Poincaré-Hopf index. This index also characterizes the cuspidal foliations having the same process of singularity reduction that the union of its separatrices. Finally we give necessary and sufficient conditions when these cuspidal foliations are generalized curves, and a characterization when they have only one separatrix.
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    Number of homogeneous components of counterexamples to the Dixmier conjecture
    (Taylor and Francis Ltd., 2025)
    Assume that P and Q are elements of A1 satisfying [𝑃,𝑄]=1. The Dixmier Conjecture for A1 says that they always generate A1. We show that if P is a sum of not more than 4 homogeneous elements of A1 then P and Q generate A1, which generalizes the main result in [Citation10].
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