We develop a theory of arithmetic Newton polygons of higher\ud order, that provides the factorization of a separable polynomial over a p-adic\ud eld, together with relevant arithmetic information about the elds generated\ud by the irreducible factors. This carries out a program suggested by . Ore.\ud As an application, we obtain fast algorithms to compute discriminants, prime\ud ideal decomposition and integral bases of number elds.Postprint (author’s final draft

We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a p-adic factorization method based on Newton polygons of higher order. The running-time and memory requirements of the algorithm appear to be very good: for a given prime number p, it computes the p-valuation of the discriminant and the factorization of p in a number field of degree 1000 in a few seconds, in a personal computer.

No abstract

We give a complete answer to the question of which polynomials occur as the characteristic polynomials of Frobenius for genus-2 curves over finite 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

Abstract. Let K be a field equipped with a discrete valuation v. In a pioneering work, MacLane determined all valuations on K(x) extending v. His work was recently reviewed and generalized by Vaquié, by using the graded algebra of a valuation. We extend Vaquié's approach by studying residual ideals of the graded algebra as an abstract counterpart of certain residual polynomials which play a key role in the computational applications of the theory. As a consequence, we determine the structure of the graded algebra of the discrete valuations on K(x) and we show how these valuations may be used to parameterize irreducible polynomials over local fields up to Okutsu equivalence.

Abstract. Let A be a Dedekind domain whose field of fractions K is a global field. Let p be a non-zero prime ideal of A, and Kp the completion of K at p. The Montes algorithm factorizes a monic irreducible separable polynomial f (x) ∈ A[x] over Kp, and it provides essential arithmetic information about the finite extensions of Kp determined by the different irreducible factors. In particular, it can be used to compute a p-integral basis of the extension of K determined by f (x). In this paper we present a new and faster method to compute p-integral bases, based on the use of the quotients of certain divisions with remainder of f (x) that occur along the flow of the Montes algorithm.

In this paper, based on techniques of Newton polygons, a result which allows the computation of a p integral basis of every quartic number field is given. For each prime integer p, this result allows to compute a p-integral basis of a quartic number field K defined by an irreducible polynomial P (X) = X 4 + aX + b ∈ Z[X] in methodical and complete generality.