Genetics of polynomials over local fields

Let $(K,v)$ be a discrete valued field with valuation ring \oo, and let \oo_v be the completion of \oo with respect to the $v$-adic topology. In this paper we discuss the advantages of manipulating polynomials in \oo_v[x] in a computer by means of OM representations of prime (monic and irreducible) polynomials. An OM representation supports discrete data characterizing the Okutsu equivalence class of the prime polynomial. These discrete parameters are a kind of DNA sequence common to all individuals in the same Okutsu class, and they contain relevant arithmetic information about the polynomial and the extension of $K_v$ that it determines.Comment: revised according to suggestions by a refere

Computation of Integral Bases

Let $A$ be a Dedekind domain, $K$ the fraction field of $A$, and $f\in A[x]$ a monic irreducible separable polynomial. For a given non-zero prime ideal $\mathfrak{p}$ of $A$ we present in this paper a new method to compute a $\mathfrak{p}$-integral basis of the extension of $K$ determined by $f$. Our method is based on the use of simple multipliers that can be constructed with the data that occurs along the flow of the Montes Algorithm. Our construction of a $\mathfrak{p}$-integral basis is significantly faster than the similar approach from $[7]$ and provides in many cases a priori a triangular basis.Comment: 22 pages, 4 figure

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

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.Comment: 51 pages. Title modified. Slight modifications in Definition 5 and Proposition 1

Single-factor lifting and factorization of polynomials over local fields

Let f (x) be a separable polynomial over a local field. The Montes algorithm computes certain approximations to the different irreducible factors of f (x), with strong arithmetic properties. In this paper, we develop an algorithm to improve any one of these approximations, till a prescribed precision is attained. The most natural application of this ‘‘single-factor lifting’’ routine is to combine it with the Montes algorithm to provide a fast polynomial factorization algorithm. Moreover, the single-factor lifting algorithm may be applied as well to accelerate the computational resolution of several global arithmetic problems in which the improvement of an approximation to a single local irreducible factor of a polynomial is requiredPostprint (published version