Economics and Business

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 the defining equation $f(x)$. 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 Montes algorithm, which is an extremely fast procedure to construct the prime ideals.

Convex cooperative games were first introduced by Shapley (1971) while population monotonic allocation schemes (PMAS) were subsequently proposed by Sprumont (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.

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 $K_p$ the completion of $K$ at $p$. The Montes algorithm factorizes a monic irreducible separable polynomial $f(x)\in A[x]$ over $K_p$, and it provides essential arithmetic information about the finite extensions of $K_p$ 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.

We generalize the well-known result of the coincidence of the bargaining set and the core for convex games (Maschler et al., 1972) to the class of games named almost-convex games, that is, coalitional games where all proper subgames are convex.

Let $(K,v)$ be a discrete valued field and let $x$ be an indeterminate. In 1936, MacLane determined all valuations on $K(x)$ extending $v$. His work has been 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 having a key role in the computational applications of the theory. This enables us to determine the structure of the graded algebra of the discrete valuations on $K(x)$. Also, let $P$ be the set of monic irreducible polynomials with coefficients in the completion $O_v$ of the valuation ring of $v$. The constructive methods of the paper yield a canonical bijection $P/≈→M$, between the set of Okutsu equivalence classes of prime polynomials and a certain MacLane space $M$.

Let $p$ be a prime number. In this paper we use an old technique of Ø. 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.

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 integral bases of 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.

Cooperative games with large core were introduced by Sharkey (1982) and the concept of Population Monotonic Allocation Scheme was defined by Sprumont (1990). Inspired by these two concepts, Moulin (1990) introduced the notion of large monotonic core giving a characterization for three-player games. In this paper we prove that all games with large monotonic core are convex. We give an effective criterion to determine whether a game has a large monotonic core and, as a consequence, we obtain a characterization for the four-player case.

We introduce our package +Ideals for Magma, designed to perform the basic tasks related to ideals in number fields without pre-computing integral bases. It is based on Montes algorithm and a number of local techniques that we have developed in a series of papers in the last years.

Let $K$ be a local field of characteristic zero, $O$ its ring of integers and $F(x)\in O[x]$ a monic irreducible polynomial. K. Okutsu attached to $F(x)$ certain primitive divisor polynomials $F_1(x),...,F_r(x)\in O[x]$, that are specially close to $F(x)$ with respect to their degree (Proc. Japan Acad. 58, 1982, Ser. A, 47–49, 87–89). In this paper we characterize the Okutsu families $F_1,...,F_r$, in terms of Newton polygons of higher order, and we derive some applications: closed formulas for certain Okutsu invariants, the discovery of new Okutsu invariants, or the construction of Montes approximations to $F(x)$; these are monic irreducible polynomials sufficiently close to $F(x)$ to share all its Okutsu invariants. This perspective widens the scope of applications of Montes algorithm (Trans. Amer. Math. Soc. 364, 2012, 361-416), (J. Théorie des Nombres de Bordeaux 23, 2011, 667-696), which can be reinterpreted as a tool to compute the Okutsu polynomials and a Montes approximation, for each irreducible factor of a monic separable polynomial $f(x)\in O[x]$.

The monotonic core of a cooperative game with transferable utility is the set formed by all its Population Monotonic Allocation Schemes (Sprumont, 1990). 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.

Ø. 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 Newton's polygon techniques, worked only when certain polynomials $f_S(Y)$, 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 $f_S(Y)$ play then an essential role.

The Chinese economy grew by 9.2% in 2011, a robust pace of growth that, even though it is the lowest in the last decade, shows that China’s economy is not going to stop growing, as some reports have suggested. Instead, China is heading for a slow landing. The reasons for its downturn lie in the slow recovery of exports due to a lower demand of the global economy, plus a decrease in investment.

According to most of the surveys consulted, for instance, those of La Caixa , the risks of a recession are very limited. In the first place, China still has enormous potential to increase its internal demand. Even though it is true that certain consumer goods are approaching saturation (mobile phones, televisions or home appliances), particularly in urban areas, other sectors, such as cars, still have a very long way to go. Regarding investment, the factors that suggest a decreasing trend are very limited, and centre on infrastructures and building. The former was seriously affected by the high-speed-train accident that took place in Wenzhou last summer. The latter is distressed due to credit restrictions being applied in order to stop the growth of inflation. However, bearing in mind that more than 300 million people in rural areas are willing to move into cities, the demand for housing is ensured. The activity indicators such as industrial production and electric production are still growing strongly (by 14% and 11% respectively in the third quarter of 2011). Finally, foreign trade, both exports and imports, are on the increase, even though not as strongly as in the past decade. Therefore, it can be concluded that the Chinese economy is not going to crash, but to cool down slightly.