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

## TL;DR

This paper introduces a quasi-linear complexity irreducibility test for polynomials in K[[x]][y], utilizing approximate roots, extending classical criteria to broader polynomial classes, and providing insights into equi-singularity types.

## Contribution

It generalizes Abhyankhar's irreducibility criterion to non algebraically closed residue fields and introduces a pseudo-irreducibility test with applications to equisingularity analysis.

## Key findings

- Quasi-linear complexity irreducibility test for K[[x]][y] polynomials.
- Extension of irreducibility criteria to non algebraically closed residue fields.
- Algorithm computes discriminant valuation and equisingularity types for pseudo-irreducible polynomials.

## Abstract

We provide an irreducibility test in the ring K[[x]][y] whose complexity is quasi-linear with respect to the valuation of the discriminant, 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 valuation of the discriminant and the equisingularity types of the germs of plane curve defined by F along the fiber x=0.