A quasi-linear irreducibility test in $$\mathbb{K}[[x]][y]$$

Abstract

We provide an irreducibility test in the ring $$\mathbb{K}[[x]][y]$$ whose complexity is quasi-linear with respect to the discriminant valuation, assuming the input polynomial $$F$$ is square-free and $$\mathbb{K}$$ is a perfect field of characteristic not dividing $$\deg(F)$$ . The algorithm uses the theory of approximate roots and may be seen as a generalisation of Abhyankar's irreducibility criterion to the case of non-algebraically closed residue fields.