Restricted Power Series Research Articles

Analytic Nullstellensätze and the model theory of valued fields

AbstractWe present a uniform framework for establishing Nullstellensätze for power series rings using quantifier elimination results for valued fields. As an application, we obtain Nullstellensätze for ‐adic power series (both formal and convergent) analogous to Rückert's complex and Risler's real Nullstellensatz, as well as a ‐adic analytic version of Hilbert's 17th Problem. Analogous statements for restricted power series, both real and ‐adic, are also considered.

Uniform Properties of Ideals in Rings of Restricted Power Series

AbstractWhen is an ideal of a ring radical or prime? By examining its generators, one may in many cases definably and uniformly test the ideal’s properties. We seek to establish such definable formulas in rings of p-adic power series, such as $\mathbb Q_{p}\langle X\rangle $ , $\mathbb Z_{p}\langle X\rangle $ , and related rings of power series over more general valuation rings and their fraction fields. We obtain a definable, uniform test for radicality, and, in the one-dimensional case, for primality. This builds upon the techniques stemming from the proof of the quantifier elimination results for the analytic theory of the p-adic integers by Denef and van den Dries, and the linear algebra methods of Hermann and Seidenberg.Abstract prepared by Madeline G. Barnicle.E-mail: barnicle@math.ucla.eduURL: https://escholarship.org/uc/item/6t02q9s4

Hensel's lemma and the intermediate value theorem over a non-Archimedean field

This paper proves that all power series over a maximal ordered Cauchy complete non-Archimedean field satisfy the intermediate value theorem on every closed interval. Hensel's lemma for restricted power series is the main tool of the proof.

On p-adic analytic continuation with applications to generating elements

AbstractGiven a prime number p and the Galois orbit O(T) of an integral transcendental element T of , the topological completion of the algebraic closure of the field of p-adic numbers, we study the p-adic analytic continuation around O(T) of functions defined by limits of sequences of restricted power series with p-adic integer coefficients. We also investigate applications to generating elements for or for some classes of closed subfields of .

Locally analytic functions in a local field of positive characteristic

In a recent paper (Buium et al., 2011 [3] ), Buium et al. proved that f is a locally analytic function from the p -adic integers, Z p to itself if and only if it is written as a restricted power series over Z p with finitely many iterates of the Fermat quotient operator. In this paper we establish the function field analogs of their result. In addition, we deduce Wagnerʼs result, which is the function field analog of Mahlerʼs result for continuous functions on Z p .

Arithmetic differential operators on [formula omitted

Given a prime p, we let δ x = ( x − x p ) / p be the Fermat quotient operator over Z p . We prove that a function f : Z p → Z p is analytic if, and only if, there exists m such that f can be represented as f ( x ) = F ( x , δ x , … , δ m x ) , where F is a restricted power series with Z p -coefficients in m + 1 variables.

On certain character sums over p ‐adic rings and their L ‐functions

AbstractIn this paper we present a new method for evaluating exponential sums associated to a restricted power series in one variable modulo pl , a power of a prime. We show that for sufficiently large l, these sums can be expressed in terms of Gauss sums. Moreover, we study the associated L ‐functions; we show that they are rational, then we determine their degrees and the weights as Weil numbers of their reciprocal roots and poles. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Multivariate Igusa theory: decay rates of exponential sums

We obtain general estimates for exponential integrals of the form Ef(y)=∫ℤpnψ(∑j=1ryjfj(x))|dx|⁠, where the fj are restricted power series over ℚp, yj ε ℚp and ψ is a nontrivial additive character on ℚp. We prove that if (f1,…,fr) is a dominant map, then |Ef(y)|<c|y|α for some c > 0 and α < 0, uniform in y, where |y|=max(|yi|)i⁠. In fact, we obtain similar estimates for a much bigger class of exponential integrals. To prove these estimates we introduce a new method to study exponential sums, namely, we use the theory of p-adic subanalytic sets and p-adic integration techniques based on p-adic cell decomposition. We compare our results with some elementarily obtained explicit bounds for Ef with fj polynomials.

A Stability Property of the Linear Growth Birth and Death Process Under Restricted Power Series Distributions

ABSTRACT : Recently Ycart has investigated a type of stability property of the linear growth Birth and Death Process according to which if the initial distribution is binomial or Poisson then it remains the same at any subsequent point of time. We have generalised this stability property under the wider class of restricted power series distributions.

A-endomorphismes de A{X}

The idea to write this paper was the completion of the paper of N. Sankaran (cf.[6]). In fact, in his theorem, he suppose that all automorphisms of the restricted power series rings with respect to the m-adic topology of the maximal ideal of a local ring is anautomorphism by substitution. But this is true for all injective endomorphisms (see our Proposition 1.5.). We study here different classes of endomorphisms of the restricted power series rings with respect to an arbitrary non nilpotent ideal in a commutative ring. To do this, we follow very closely the techniques of M.J. O'Malley concerning the rings of power series, but adapted to the case of the rings of restricted power series (cf.[3],[4]).

Automorphisms of restricted power series rings

Automorphisms of restricted power series rings