Algebraic Formal Power Series Research Articles

On Mahler's Classification of Formal Power Series Over a Finite Field

Abstract Let K be a finite field, K(x) be the field of rational functions in x over K and K be the field of formal power series over K. We show that under certain conditions integral combinations with algebraic formal power series coefficients of a U 1-number in K are Um -numbers in K , where m is the degree of the algebraic extension of K(x), determined by these algebraic formal power series coefficients.

ON TRANSCENDENTAL CONTINUED FRACTIONS IN FIELDS OF FORMAL POWER SERIES OVER FINITE FIELDS

AbstractIn the field of formal power series over a finite field, we prove a result which enables us to construct explicit examples of $U_{m}$ -numbers by using continued fraction expansions of algebraic formal power series of degree $m>1$ .

On certain recurrent and automatic sequences in finite fields

In this work we consider the general question: for a given algebraic formal power series with coefficients in a finite field, what kind of regularity (if any) can be expected for the partial quotients of the above power series in continued fraction expansion? Such a question is natural, since by a theorem of Christol, the coefficients of an algebraic power series over a finite field form an automatic sequence. Certain algebraic continued fractions are such that the sequence of the leading coefficients of the partial quotients is automatic. Here we give a rather general family of such sequences. Moreover, inspired by these examples, we give two criteria on automatic sequences, which allow us to obtain two new families of automatic sequences in an arbitrary finite field.

CONTINUED FRACTION AND DIOPHANTINE EQUATION

Abstract. Our paper is devoted to the study of certain diophantineequations on the ring of polynomials over a finite field, which are inti-mately related to algebraic formal power series which have partial quo-tients of unbounded degree in their continued fraction expansion. Inparticular it is shown that there are Pisot formal power series with de-gree greater than 2, having infinitely many large partial quotients in theirsimple continued fraction expansions. This generalizes an earlier resultof Baum and Sweet for algebraic formal power series. 1. IntroductionLet F q be a finite field of characteristic p with q elements. We considerF q [X], F q (X) and F q ((X −1 )) as analogues of Z, Q and R respectively. Ifw =P +∞n=n 0 a n X −n is an element of F q ((X −1 )) with a n 0 6= 0, we introducethe absolute value |w| = q −n 0 and |0| = 0. Diophantine approximation inthe function field case was initiated by K. Mahler [10]. In the case of realnumbers, it is well known that Liouville’s theorem was the beginning of along path, with the works of Thue, Siegel, Dyson and others, leading of thecelebrated Roth’s theorem which was established in 1955. This improvementcan be transposed in fields of power series if the base field has characteristiczero, as shown by Uchiyama in 1960. But this is not the case in positivecharacteristicand consequentlythe studyofrationalapproximationtoalgebraicelements becomes somewhat more complex.Many examples can be studied. A special case is the one where the algebraicelement w satisfies an equation of the form w =

Diophantine Approximation for Certain Algebraic Formal Power Series in Positive Characteristic

Abstract.In this paper, we study rational approximations for certain algebraic power series over a finite field. We obtain results for irrational elements of strictly positive degree satisfying an equation of the typewhere (A, B, C) ∊ (𝔽q[X])2 × 𝔽*q [X]. In particular, under some conditions on the polynomials A, B and C, we will give well approximated elements satisfying this equation.

Weighted Logics for Nested Words and Algebraic Formal Power Series

Nested words, a model for recursive programs proposed by Alur and Madhusudan, have recently gained much interest. In this paper we introduce quantitative extensions and study nested word series which assign to nested words elements of a semiring. We show that regular nested word series coincide with series definable in weighted logics as introduced by Droste and Gastin. For this we establish a connection between nested words and the free bisemigroup. Applying our result, we obtain characterizations of algebraic formal power series in terms of weighted logics. This generalizes results of Lautemann, Schwentick and Therien on context-free languages.

Existence and uniqueness of Tronquée solutions of the fourth-order Jimbo-Miwa second Painlevé equation

We consider the asymptotic limit as the independent variable approaches infinity, of the fourth-order second Painlevé equation obtained from a hierarchy based on the Jimbo-Miwa Lax pair. We prove that there exist two families of algebraic formal power series solutions and that there exist true solutions with these behaviours in sectors σ \sigma of the complex plane. Given σ \sigma we also prove that there exists a wider sector Σ ⊃ σ \Sigma \supset \sigma in which there exists a unique solution in each family. These provide the analogue of Boutroux’s tri-tronquée solutions for the classical second Painlevé equation. Surprisingly, they also extend beyond the tri-tronquée solutions in the sense that we find penta-, hepta-, ennea-, and hendeca-tronquée solutions.

Transcendental continued fractions over K_p(X)

The purpose behind this work is to construct from a family of algebraic formal power series of degree more than 2, a family of transcendental fractions over IKp(X).

Un critére de transcendance de fractions continues dans [formula omitted

There are uncountably many continued fractions of formal power series with bounded sequence of partial quotients, most of them are transcendental, but it is not very easy to give examples. This paper creates many families of such examples in an interesting way using the continued fractions of algebraic series with bounded partial quotients (none such are known and presumably do not exist for real numbers), given by Baum-Sweet, Mills-Robbins, Lasjaunias [continued fractions for certain algebraic formal power series over a finite field, Proceedings of the Sixth International Conference on Finite Fields and Applications, Oaxaca, Mexico, May 2001, Springer, Berlin, pp. 220–228], and recent many families by Thakur [J. Number Theory 59 (1996) 248–261; J. Number Theory 66 (1997) 129–147; J. Number Theory 79 (1999) 284–291].

Continued Fractions for Algebraic Formal Power Series over a Finite Base Field

We consider the continued fraction expansion of certain algebraic formal power series when the base field is finite. We are concerned with the property of the sequence of partial quotients being bounded or unbounded. We formalize the approach introduced by Baum and Sweet (1976), which applies to the elements of a particular subset of algebraic power series. We illustrate this method with a result when the base field is F2.

A computational model for algebraic power series

Up to now, no computational model is known to perform effective commutative algebra for ideals in a computable ring of formal power series, while the theory for this is quite developed at least since [7]. The particular case of algebraic formal power series comes out naturally, when studying singular points of algebraic varieties, for instance, in the NewtonPuiseux algorithm for determining the analytic branches of a curve at a singular point and, more generally, when studying analytic components of a complex algebraic variety. We propose here to develop a computational model for algebraic formal power series, already introduced in [l], based on a symbolic codification of the series by means of the Implicit Function Theorem, i.e., we will consider algebraic series as the unique solutions of suitable functional equations, which we call Locally Smooth Systems. We then reduce the problem of handling a finite set of algebraic series to some corresponding problem involving suitable polynomial rings. In this model we will show that most of the usual local commutative algebra can be effectively performed on algebraic series, since we can reduce to the polyno-

La queste del saint Gr a(AL): A computational approach to local algebra

We show how, by means of the Tangent Cone Algorithm, the basic functions related to the maximal ideal topology of a local ring can be effectively computed in the situations of geometrical significance, i.e.: (1) localizations of coordinate rings of a variety at the prime ideal defining a subvariety, (2) rings of algebraic formal power series rings. In particular we show how the method of “associated graded rings” can be turned into an effective tool to compute local algebraic invariants of varieties.

Ranking and formal power series

We prove that the ranking problem for unambiguous context-free languages is NC 1-reducible to the value problem for algebraic formal power series in noncommuting variables. In the particular case of regular languages we show that the problem of ranking is NC 1-reducible to the problem of counting the number of strings of given length in suitable regular languages. As a consequence ranking problems for regular languages are NC 1-reducible to integer division and hence computable by log-space uniform boolean circuits of polynomial size and depth O(log n log log n), or by P-uniform boolean circuits of polynomial size and depth O(log n).

Polyominoes and animals: Some recent results

We give a survey of recent works relating algebraic languages and formal power series with the enumeration of polyominoes (and animals). More precisely, encoding these structures with words yield new exact results.

Counting problems and algebraic formal power series in noncommuting variables

In this work we study the complexity of certain counting functions related to formal power series in noncommuting variables. We prove that, for every algebraic formal power series in Z 《 ∑》, the problem of computing the corresponding counting function is NC 1-reducible to integer division. As a consequence, for every unambiguous context-free language L⊆ ∑ *, the problem of computing #{ xϵ L:| x|= n} is also NC 1-reducible to integer division. Therefore all t hese counting problems are solvable by families of log-space uniform boolean circuits of depth O(log n log log n) and polynomial size.

Cyclic derivation of noncommutative algebraic power series

We show that the cyclic derivative of any algebraic formal power series in noncommuting variables is again algebraic.