Jesús Montes

A New Computational Approach to Ideal Theory in Number Fields

Let K be the number field determined by a monic irreducible polynomial f(x) with integer coefficients. In previous papers we parameterized the prime ideals of K in terms of certain invariants attached to Newton polygons of higher order of f(x). In this paper we show how to carry out the basic operations on fractional ideals of K in terms of these constructive representations of the prime ideals. From a computational perspective, these results facilitate the manipulation of fractional ideals of K avoiding two heavy tasks: the construction of the maximal order of K and the factorization of the discriminant of f(x). The main computational ingredient is the Montes algorithm, which is an extremely fast procedure to construct the prime ideals.

On a theorem of Ore

Abstract O. Ore (Math. Ann. 99, 1928, 84–117) developed a method for obtaining the absolute discriminant and the prime-ideal decomposition of the rational primes in a number field K. The method, based on Newtons polygon techniques, worked only when certain polynomials ƒ S (Y) , attached to any side S of the polygon, had no multiple factors. These results are generalized in this paper finding a much weaker condition, effectively computable, under which it is still possible to give a complete answer to the above questions. The multiplicities of the irreducible factors of the polynomials ƒ S (Y) play then an essential role.

NEWTON POLYGONS AND p-INTEGRAL BASES OF QUARTIC NUMBER FIELDS

Let p be a prime number. In this paper we use an old technique of O. Ore, based on Newton polygons, to construct in an efficient way p-integral bases of number fields defined by p-regular equations. To illustrate the potential applications of this construction, we derive from this result an explicit description of a p-integral basis of an arbitrary quartic field in terms of a defining equation.

A NOTE ON THE MONOTONIC CORE

The monotonic core of a cooperative game with transferable utility is the set formed by all its Population Monotonic Allocation Schemes. In this paper we show that this set always coincides with the core of a certain game, with and without restricted cooperation, associated to the initial game.

A note: characterizations of convex games by means of population monotonic allocation schemes

Convex cooperative games were first introduced by Shapley (Int J Game Theory 1:11–26, 1971) while population monotonic allocation schemes (PMAS) were subsequently proposed by Sprumont (Games Econ Behav 2:378–394, 1990). In this paper we provide several characterizations of convex games and introduce three new notions: PMAS-extendability, PMAS-exactness, and population monotonic set schemes, which imitate the classical concepts that they extend. We show that all of these notions provide new characterizations of the convexity of the game.