18 research outputs found
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 ââsingle-factor liftingââ 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
Computation of Integral Bases
Let be a Dedekind domain, the fraction field of , and
a monic irreducible separable polynomial. For a given non-zero prime ideal
of we present in this paper a new method to compute a
-integral basis of the extension of determined by . 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 -integral basis is significantly faster than the similar
approach from and provides in many cases a priori a triangular basis.Comment: 22 pages, 4 figure
On the equivalence of types
Types over a discrete valued field are computational objects that
parameterize certain families of monic irreducible polynomials in ,
where is the completion of at . Two types are considered to be
equivalent if they encode the same family of prime polynomials. In this paper,
we characterize the equivalence of types in terms of certain data supported by
them
Okutsu-montes representations of prime ideals of one-dimensional integral closures
This is a survey on Okutsu-Montes representations of prime ideals of certain one-dimensional integral closures. These representations facilitate the computational resolution of several arithmetic tasks concerning prime ideals of global fields
Triangular bases of integral closures
In this work, we consider the problem of computing triangular bases of
integral closures of one-dimensional local rings.
Let be a discrete valued field with valuation ring and
let be the maximal ideal. We take , a
monic irreducible polynomial of degree and consider the extension as well as the integral closure of
in , which we suppose to be finitely generated as an -module.
The algorithm , presented in this paper, computes
triangular bases of fractional ideals of . The theoretical
complexity is equivalent to current state of the art methods and in practice is
almost always faster. It is also considerably faster than the routines found in
standard computer algebra systems, excepting some cases involving very small
field extensions