A quasi-linear irreducibility test in K[[x]][y]

TL;DR

This paper introduces a quasi-linear complexity irreducibility test for polynomials in K[[x]][y], extending existing criteria to non algebraically closed residue fields using approximate roots.

Contribution

It presents a new irreducibility test with quasi-linear complexity in the discriminant valuation, generalizing Abhyankhar's criterion for broader residue fields.

Findings

Test has quasi-linear complexity in discriminant valuation

Applicable to square-free polynomials in K[[x]][y]

Works over perfect fields with characteristic constraints

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 generalisation of Abhyankhar's irreducibility criterion to the case of non algebraically closed residue fields.