On Kummer extensions of the power series field

In this paper we study the Kummer extensions K ′ of a power series field K = k ((X₁, …, X_r)), where k is an algebraically closed field of arbitrary characteristic, with special emphasis in the case where K ′ is generated by a Puiseux power series. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finiteness results concerning algebraic power series

We construct an explicit filtration of the ring of algebraic power series by constructible sets, measuring the complexity of these series. As an example of use of this, we give a bound on the dimension of the set of algebraic power series of bounded complexity lying on an algebraic variety defined over the field of power series.     

On the algebraic closure in rings

The ``algebraic closure" of a subset of a ring is an algebraic analogue of topological closure.

     

Absolute integral closure in positive characteristic

Let R be a local Noetherian domain of positive characteristic. A theorem of Hochster and Huneke [M. Hochster, C. Huneke, Infinite integral extensions and big Cohen–Macaulay algebras, Ann. of Math. 135 (1992) 53–89] states that if R is excellent, then the absolute integral closure of R is a big Cohen–Macaulay algebra. We prove that if R is the homomorphic image of a Gorenstein local ring, then all the local cohomology (below the dimension) of such a ring maps to zero in a finite extension of the ring. As a result there follow an extension of the original result of Hochster and Huneke to the case in which R is a homomorphic image of a Gorenstein local ring, and a considerably simpler proof of this result in the cases where the assumptions overlap, e.g., for complete Noetherian local domains.     

On vanishing coefficients of algebraic power series over fields of positive characteristic

Let K be a field of characteristic p>0 and let f(t ₁,…,t _d ) be a power series in d variables with coefficients in K that is algebraic over the field of multivariate rational functions K(t ₁,…,t _d ). We prove a generalization of both Derksen’s recent analogue of the Skolem–Mahler–Lech theorem in positive characteristic and a classical theorem of Christol, by showing that the set of indices (n ₁,…,n _d )∈? ^d for which the coefficient of \(t_{1}^{n_{1}}\cdots t_{d}^{n_{d}}\) in f(t ₁,…,t _d ) is zero is a p-automatic set. Applying this result to multivariate rational functions leads to interesting effective results concerning some Diophantine equations related to S-unit equations and more generally to the Mordell–Lang Theorem over fields of positive characteristic.     

The real field with convergent generalized power series

We construct a model complete and o-minimal expansion of the field of real numbers in which each real function given on by a series with and for some is definable. This expansion is polynomially bounded.

     

Unique factorization in generalized power series rings

Let be a field of characteristic zero and let denote the ring of generalized power series (i.e., formal sums with well-ordered support) with coefficients in , and non-positive real exponents. Berarducci (2000) constructed an irreducible omnific integer, in the sense of Conway (2001), by first proving that an element of that is not divisible by a monomial and whose support has order type (or for some ordinal ) must be irreducible. In this paper, we consider elements of with support of order type . The irreducibility of these elements cannot be deduced solely from the order type of their support and, after developing new tools for studying these elements, we exhibit both reducible and irreducible elements of this type. We further prove that all elements whose support has order type and which are not divisible by a monomial factor uniquely into irreducibles. This provides, in the ring , a class of reducible elements for which we have unique factorization into irreducibles.

     

Surjectivity of power maps of real algebraic groups

In this paper we study the surjectivity of the power maps g?gn for real points of algebraic groups defined over reals. The results are also applied to study the exponentiality of such groups.     

A new summation method for power series with rational coefficients

We show that an asymptotic summation method, recently proposed by the authors, can be conveniently applied to slowly convergent power series whose coefficients are rational functions of the summation index. Several numerical examples are presented.

     

Factorization in generalized power series

The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): . Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway's series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If we can give the following test for irreducibility based only on the order type of the support of the series: if the order type is either or of the form and the series is not divisible by any monomial, then it is irreducible. To handle the general case we use a suggestion of
M.-H. Mourgues, based on an idea of Gonshor, which allows us to reduce to the special case . In the final part of the paper we study the irreducibility of series with finite support.

     

Continued fractions for hyperquadratic power series over a finite field

An irrational power series over a finite field of characteristic p is called hyperquadratic if it satisfies an algebraic equation of the form x=(Ax^r+B)/(Cx^r+D), where r is a power of p and the coefficients belong to . These algebraic power series are analogues of quadratic real numbers. This analogy makes their continued fraction expansions specific as in the classical case, but more sophisticated. Here we present a general result on the way some of these expansions are generated. We apply it to describe several families of expansions having a regular pattern.     

Uniserial rings of skew generalized power series

Let R be a ring, S a strictly ordered monoid and a monoid homomorphism. In this paper we obtain some necessary conditions for the skew generalized power series ring RS,ω to be right (respectively left) uniserial, and we prove that these conditions are also sufficient when the monoid S is commutative or totally ordered.     

On the integral dependence of power series rings

We prove the following results: (1) Let R ? S be two commutative rings. Suppose that dim R = 0.If f(X) ∈ S[[X]]is integral over R[[X]], then every coefficient of f(X) is integral over R. (2) Let dim R ≥ 1. There exists a ring S containing R and a power series f(X) ∈ S[[X]]such that f(X) is integral over R[[X]], but not all coefficients of f(X) are integral over R. (3) Let k ? R. Suppose that R is algebraic over the field k. Then R[[X]] is integral over k[[X]] if and only if the nilradical of R is nilpotent and the separable degree and the inseparable exponent of R _red over k are finite.     

Transcendence of some multivariate power series

We prove some transcendence results for the sums of some multivariate serms of the form ∑j1,j2,...,jm=0 ^∞Cj1j2...jm（r1^j1r2^j2...rm^jm） for n = 1, 2, where Cj1j2...jm are some rational functions of j1 ＋ j2 ＋ ... ＋ jm.