T-functions revisited: new criteria for bijectivity/transitivity

The paper presents new criteria for bijectivity/transitivity of T-functions and a fast knapsack-like algorithm of evaluation of a T-function. Our approach is based on non-Archimedean ergodic theory: Both the criteria and algorithm use van der Put series to represent 1-Lipschitz p-adic functions and to study measure-preservation/ergodicity of these.     

Automata finiteness criterion in terms of van der Put series of automata functions

In the paper we develop the p-adic theory of discrete automata. Every automaton \mathfrakA\mathfrak{A} (transducer) whose input/output alphabets consist of p symbols can be associated to a continuous (in fact, 1-Lipschitz) map from p-adic integers to p-adic integers, the automaton function f_\mathfrakA f_\mathfrak{A} . The p-adic theory (in particular, the p-adic ergodic theory) turned out to be very efficient in a study of properties of automata expressed via properties of automata functions. In the paper we prove a criterion for finiteness of the number of states of automaton in terms of van der Put series of the automaton function. The criterion displays connections between p-adic analysis and the theory of automata sequences.     

Van der Put basis and <Emphasis Type="Italic">p</Emphasis>-adic dynamics

We consider a function g: Z _p → Z _p and its the van der Put series. Then we get a criteria of Haar’s measure preserving compatible p-adic functions which, actually, need not be uniformly differentiable modulo p. This is used to study ergodicity of p-adic dynamical systems [2, 16].     

On ergodicity of p-adic dynamical systems for arbitrary prime p

In this paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of 1-Lipschitz p-adic functions that are defined on (and valuated in) the space ? _p of p-adic integers for any prime p. The conditions are stated in terms of coordinate representations of p-adic functions.     

On measure-preserving functions over ?3

This paper is devoted to (discrete) p-adic dynamical systems, an important domain of algebraic and arithmetic dynamics [31]?C[41], [5]?C[8]. In this note we study properties of measurepreserving dynamical systems in the case p = 3. This case differs crucially from the case p = 2. The latter was studied in the very detail in [43]. We state results on all compatible functions which preserve measure on the space of 3-adic integers, using previous work of A. Khrennikov and author of present paper, see [24]. To illustrate one of the obtained theorems we describe conditions for the 3-adic generalized polynomial to be measure-preserving on ?₃. The generalized polynomials with integral coefficients were studied in [17, 33] and represent an important class of T-functions. In turn, it is well known that T-functions are well-used to create secure and efficient stream ciphers, pseudorandom number generators.     

Behaviour of Hensel perturbations of p-adic monomial dynamical systems

In the framework of non-Archimedean (p-adic) analysis we study cyclic behaviour of polynomial discrete dynamical systems (iterations of polynomial maps). One of the main tools of our investigation is Hensel's lemma (a p-adic analogue of Newton's method). Our considerations will lead to formulas for the number cycles of a specific length and for the total number of cycles. We will also study the distribution of cycles in the different p-adic fields.     

Measure-Preservation and the Existence of a Root of <Emphasis Type="Italic">p</Emphasis>-Adic 1-Lipschitz Functions in Mahler’s Expansion

In this paper, we provide necessary and sufficient conditions for 1-Lipschitz functions that are uniformly differentiable mod p on ?_p to be measure-preserving, in Mahler’s expansion, and show that these conditions can be modified to guarantee the existence of a root of p-adic Lipschitz functions.     

On a class of rational p-adic dynamical systems

In this paper we investigate the behavior of trajectories of one class of rational p-adic dynamical systems in complex p-adic field Cp. We studied Siegel disks and attractors of such dynamical systems. We found the basin of the attractor of the system. It is proved that such dynamical systems are not ergodic on a unit sphere with respect to the Haar measure.     

The pointwise convergence of p-adic Mbius maps

The convergence of linear fractional transformations is an important topic in mathematics.We study the pointwise convergence of p-adic Mbius maps,and classify the possibilities of limits of pointwise convergent sequences of Mbius maps acting on the projective line P1(C p),where C p is the completion of the algebraic closure of Q p.We show that if the set of pointwise convergence of a sequence of p-adic Mbius maps contains at least three points,the sequence of p-adic Mbius maps either converges to a p-adic Mbius map on the projective line P1(C p),or converges to a constant on the set of pointwise convergence with one unique exceptional point.This result generalizes the result of Piranian and Thron(1957)to the non-archimedean settings.     

A Note on Complex <Emphasis Type="Italic">p</Emphasis>-Adic Exponential Fields

In this paper we apply Ax-Schanuel’s Theorem to the ultraproduct of p-adic fields in order to get some results towards algebraic independence of p-adic exponentials for almost all primes p.     

Gauss sums and multinomial coefficients

Consider a Gauss sum for a finite field of characteristic p, where p is an odd prime. When such a sum (or a product of such sums) is a p-adic integer we show how it can be realized as a p-adic limit of a sequence of multinomial coefficients. As an application we generalize some congruences of Hahn and Lee to exhibit p-adic limit formulae, in terms of multinomial coefficients, for certain algebraic integers in imaginary quadratic fields related to the splitting of rational primes. We also give an example illustrating how such congruences arise from a p-integral formal group law attached to the p-adic unit part of a product of Gauss sums.     

Sets of Minimality of (1 − 1)-Rational Functions

We describe dynamical systems associated to (1 ? 1)-rational functions on the field of p-adic numbers.We focus on sets of minimality of such systems.     

Computable valued fields

We investigate the computability-theoretic properties of valued fields, and in particular algebraically closed valued fields and p-adically closed valued fields. We give an effectiveness condition, related to Hensel’s lemma, on a valued field which is necessary and sufficient to extend the valuation to any algebraic extension. We show that there is a computable formally p-adic field which does not embed into any computable p-adic closure, but we give an effectiveness condition on the divisibility relation in the value group which is sufficient to find such an embedding. By checking that algebraically closed valued fields and p-adically closed valued fields of infinite transcendence degree have the Mal’cev property, we show that they have computable dimension \(\omega \).     

The algebraic functional equation of Selmer groups for CM fields

Because the analytic functional equation holds for Katz p-adic L-function for CM fields, the algebraic functional equation of the Selmer groups for CM fields is expected to hold. In this note we prove it following the specialization principle developed by T. Ochiai (2005) in [Och05].     

Denseness results in the theory of algebraic fields

We study when the property that a field is dense in its real and p-adic closures is elementary in the language of rings and deduce that all models of the theory of algebraic fields have this property.     

On congruences for the traces of powers of some matrices

In a series of recent papers, V.I. Arnold studied many questions concerning the statistics and dynamics of powers of elements in algebraic systems. In particular, on the basis of experimental data, he proposed an Euler-type congruence for the traces of powers of integer matrices as a conjecture. The proof of this conjecture was deduced from the author’s theorem (obtained at the end of 2004) on congruences for the traces of powers of elements in number fields. Recently, it turned out that there also exist other approaches to congruences for the traces of powers of integer matrices. In the present paper, the author’s results of 2004 are strengthened and a survey of their relations to number theory, theory of dynamical systems, combinatorics, and p-adic analysis is given. The main conclusion of this survey is that all approaches considered here ultimately reflect different points of view on a certain simple but important phenomenon in mathematics.     

Nonsymmetric Macdonald polynomials and matrix coefficients for unramified principal series

We show how a certain limit of the nonsymmetric Macdonald polynomials appears in the representation theory of semisimple groups over p-adic fields as matrix coefficients for the unramified principal series representations. The result is the nonsymmetric counterpart of a classical result relating the same limit of the symmetric Macdonald polynomials to zonal spherical functions on groups of p-adic type.     

On the structure of p-adic subanalytic functions and sets

We prove a p-adic version of the Lion?CRolin preparation theorem. As a consequence, we obtain a cell decomposition theorem, which can be viewed as an extension of Denef??s cell decomposition theorem to the p-adic analytic case. Using these theorems, we give shorter and more explicit proofs of some of the results by Denef and van den Dries on p-adic subanalytic sets.