We give a criterium of holomorphy for some type formal power series. This gives a stronger form of a Rothstein's type extension theorem for a particular ring of holomorphic functions. We consider the set R ⊂ C[[z 1 , z 2 ]], z 1 , z 2 ∈ C k × C m of formal power series of the form f (z) = f (z 1 , z 2) = n P n (z 2)z n 1 where P n is a polynomial in m variables of total degree d 0 P n ≤ C 0 + C...

A ring R is called a left APP-ring if the left annihilator lR(Ra) is right s-unital as an ideal of R for any element a ∈ R. We consider left APP-property of the skew formal power series ring R[[x;α]] where α is a ring automorphism of R. It is shown that if R is a ring satisfying descending chain condition on right annihilators then R[[x;α]] is left APP if and only if for any sequence (b0, b1, ....

This paper dealt with by studying continued fractions of the form c 1 1 + c 1 1 + · · · + c 1 1 + · · · Necessary and sufficient conditions are given for a sequence of it to be convergent in the formal powers series case.

In this paper, we consider nonlinear algebraic ordinary differential equations (AODEs) and study their formal power series solutions. Our method is inherited from Lemma 2.2 in [J. Denef and L. Lipshitz, Power series solutions of algebraic differential equations, Mathematische Annalen, 267(1984), 213-238] for expressing high order derivatives of a differential polynomial via their lower order on...

The aim of this paper is to use formal power series techniques to study the structure of small arithmetic complexity classes such as GapNC and GapL. More precisely, we apply the Kleene closure of languages and the formal power series operations of inversion and root extraction to these complexity classes. We define a counting version of Kleene closure and show that it is intimately related to i...

In the case of a source-free plane-layered elastic medium, and for direct rays, the formal series ‘solution’ of the elastic equations in inverse powers of frequency can be obtained to any desired order by a process of integration by parts of a suitable iterate. The second iteration of the standard wave vector must be employed to ensure that all O(w-‘) terms are included in the iterative solutio...

Linear finite transducers underlie a series of schemes for Public Key Criptography (PKC) proposed in the 90s of the last century. The uninspiring and arid language then used, condemned these works to oblivion. Although some of these schemes were, after, shown to be insecure, the promise of a new system of PKC relying in different complexity assumptions is still quite exciting. The algorithms th...

Let (an)n≥0 be a sequence of complex numbers such that its generating series satisfies ∑ n≥0 ant n = h(t) (1−t)d for some polynomial h(t). For any r ≥ 1 we study the transformation of the coefficient series of h(t) to that of h〈r〉(t) where ∑ n≥0 anrt n = h 〈r〉(t) (1−t)d . We give a precise description of this transformation and show that under some natural mild hypotheses the roots of h〈r〉(t) c...

We present the main results on 2PFA’s and on the Hadamard quotient of formal power series, the connection between the two notions being a result stating that the event defined by a 2PFA is the Hadamard quotient of two rational power series.

We consider the problem of predicting long sequences of zero coefficients in a power series obtained by multiplication, division or reversion (where all coefficients are integers). We describe efficient randomized algorithms whose probability of error can be controlled by Publisher: Please supply received and accepted dates. the user. A runtime analysis is given and some experimental results ar...