In this paper we study the geometric location of periodic points of power series defined over fields of prime characteristic @math . More specifically, we find a lower bound for the absolute value of all periodic points in the open unit disk of minimal period @math of 2-ramified power series. We prove that this bound is optimal for a large class of power series. Our main technical result is a computation of the first significant terms of the @math th iterate of 2-ramified power series. As a by-product we obtain a self-contained proof of the characterization of 2-ramified power series.
In @cite_5 the authors give a corresponding result of Theorem for minimally ramified power series, where @math is expressed in terms of the coefficients of the first two non-linear terms. Provided the information from Theorem we can make a corresponding version of Corollary for minimally ramified power series, where the conditions for optimality are expressed in terms of the four lowest degree non-linear terms.
@cite_5
2135702680
We study analytic germs in one variable with a parabolic fixed point at the origin, over an ultrametric ground field of positive characteristic. It is conjectured that for such a germ the origin is isolated as a periodic point. Our main result is an affirmative solution of this conjecture in the case of a generic germ with a prescribed multiplier. The genericity condition is explicit: the power series is minimally ramified, i.e. the degree of the first nonlinear term of each of its iterates is as small as possible. Our main technical result is a computation of the first significant terms of a minimally ramified power series. From this we obtain a lower bound for the norm of nonzero periodic points, from which we deduce our main result. As a by-product we give a new and self-contained proof of a characterization of minimally ramified power series in terms of the iterative residue. Let ( O ) k be the ring of integers of a finite extension k of the field ( Q ) p of p-adic numbers. The endomorphisms of a formal group law defined over ( O ) k provide nontrivial examples of commuting formal series with coefficients in ( O ) k . This article deals with the inverse problem formulated by Jonathan Lubin within the context of non-Archimedean dynamical systems. We present a large family of series, with coefficients in ( Z ) p , which satisfy Lubin's conjecture. These series are constructed with the help of Lubin–Tate formal group laws over ( Q ) p . We introduce the notion of minimally ramified series which turn out to be modulo p reductions of some series of this family. The commutant monoids of these minimally ramified series are determined by using the Fontaine–Wintenberger theory of the field of norms which allows an interpretation of them as automorphisms of ( Z ) p -extensions of local fields of characteristic zero. A particularly effective example illustrating the paper is given by a family of series generalizing Cebysev polynomials We develop power series approximations for a discrete-time queueing system with two parallel queues and one processor. If both queues are nonempty, a customer of queue 1 is served with probability β, and a customer of queue 2 is served with probability 1?β. If one of the queues is empty, a customer of the other queue is served with probability 1. We first describe the generating function U(z 1,z 2) of the stationary queue lengths in terms of a functional equation, and show how to solve this using the theory of boundary value problems. Then, we propose to use the same functional equation to obtain a power series for U(z 1,z 2) in β. The first coefficient of this power series corresponds to the priority case β=0, which allows for an explicit solution. All higher coefficients are expressed in terms of the priority case. Accurate approximations for the mean stationary queue lengths are obtained from combining truncated power series and Pade approximation. We recently proposed a data structure, called associative-commutative discrimination nets, that supports efficient algorithms for (many-to-one) term matching in the presence of associative-commutative functions. In this paper we discuss the integration of such discrimination nets into an actual equational theorem prover and report on corresponding experiments. The general associative-commutative matching problem is known to be NP-complete, but can be solved in polynomial time if the given terms are linear, i.e., do not contain multiple occurrences of the same variable. We therefore have implemented a two-stage matching procedure. First we check whether a match exists for the linearized versions of the given terms (where different occurrences of the same variable are replaced by different new variables). If a match for the linearized terms does exist, we then determine whether there is also a match for the original, non-linear terms (i.e., whether the proposed substitutions for different occurrences of the same variable are consistent). Our experimental results indicate that this approach works very well in theorem proving, where most matching attempts actually fail and are filtered out during the first stage, so that the second, more expensive stage of the algorithm is only needed in comparatively few cases.