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Using approximate roots for irreducibility and equi-singularity issues in K[[x]][y]

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Abstract

We provide an irreducibility test in the ring K[[x]][y] whose complexity is quasi-linear with respect to the discriminant valuation, assuming the input polynomial F square-free and K a perfect field of characteristic zero or greater than deg(F). The algorithm uses the theory of approximate roots and may be seen as a generalization of Abhyankhar's irreducibility criterion to the case of non algebraically closed residue fields. More generally, we show that we can test within the same complexity if a polynomial is pseudo-irreducible, a larger class of polynomials containing irreducible ones. If F is pseudo-irreducible, the algorithm computes also the discriminant valuation of F and the equisingularity classes of the germs of plane curves defined by F along the fiber x = 0.
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Dates and versions

hal-02137331 , version 1 (22-05-2019)
hal-02137331 , version 2 (08-11-2019)

Identifiers

  • HAL Id : hal-02137331 , version 2

Cite

Adrien Poteaux, Martin Weimann. Using approximate roots for irreducibility and equi-singularity issues in K[[x]][y]. 2019. ⟨hal-02137331v2⟩
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