Newton polygons of higher order in algebraic number theory

Newton Polygons of Higher Order in Algebraic Number Theory

We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a p-adic field, together with relevant arithmetic information about the fields generated by the irreducible factors. This carries out a program suggested by Ø. Ore. As an application, we obtain fast algorithms to compute discriminants, prime ideal decomposition and i...

Higher Order Bernoulli Polynomials and Newton Polygons

Higher Newton Polygons and Integral Bases

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 us...

Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields

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 nu...

Optimal Higher Order Delaunay Triangulations of Polygons

This paper presents an algorithm to triangulate polygons optimally using order-k Delaunay triangulations, for a number of quality measures. The algorithm uses properties of higher order Delaunay triangulations to improve the O(n) running time required for normal triangulations to O(kn log k + kn log n) expected time, where n is the number of vertices of the polygon. An extension to polygons wit...