Rationality of a certain formal power series attached to local densities of quadratic forms

We will prove the rationality of the power series attached to local densities defined by the author     

Rational approximation to formal power series

A general method for obtaining rational approximations to formal power series is defined and studied. This method is based on approximate quadrature formulas. Newton-Cotes and Gauss quadrature methods are used. It is shown that Padé approximants and the ε-algorithm are related to Gaussian formulas while linear summation processes are related to Newton-Cotes formulas. An example is exhibited which shows that Padé approximation is not always optimal. An application to e^−t is studied and a method for Laplace transform inversion is proposed.     

A remark on the Ikeda lift and local singular series polynomials

We relate a certain number-theoretic function occurring in a linear version of Ikeda's lifting map to local singular series polynomials.

     

On universal formal power series

The point source of this work is Seleznev's theorem which asserts the existence of a power series which satisfies universal approximation properties in C^∗. The paper deals with a strengthened version of this result. We establish a double approximation theorem on formal power series using a weighted backward shift operator. Moreover we give strong conditions that guarantee the existence of common universal series of an uncountable family of weighted backward shift with respect to the simultaneous approximation. Finally we obtain results on admissible growth of universal formal power series. We especially prove that you cannot control the defect of analyticity of such a series even if there exist universal series in the well-known intersection of formal Gevrey classes.     

On composite formal power series

Let be a holomorphic map from to defined in a neighborhood of such that . If the Jacobian determinant of is not identically zero, P. M. Eakin et G. A. Harris proved the following result: any formal power series such that is analytic is itself analytic. If the Jacobian determinant of is identically zero, they proved that the previous conclusion is no more true.

The authors get similar results in the case of formal power series satifying growth conditions, of Gevrey type for instance. Moreover, the proofs here give, in the analytic case, a control of the radius of convergence of by the radius of convergence of .

RÉSUMÉ. Soit une application holomorphe de dans définie dans un voisinage de et vérifiant . Si le jacobien de n'est pas identiquement nul au voisinage de , P.M. Eakin et G.A. Harris ont établi le résultat suivant: toute série formelle telle que est analytique est elle-même analytique. Si le jacobien de est identiquement nul, ils montrent que la conclusion précédente est fausse.

Les auteurs obtiennent des résultats analogues pour les séries formelles à croissance contrôlée, du type Gevrey par exemple. De plus, les preuves données ici permettent, dans le cas analytique, un contrôle du rayon de convergence de par celui de .

     

t-closed rings of formal power series

Let A ì BA\subset B be rings. We say that A is t-closed in B, if for each a ? Aa\in A and b ? Bb\in B such that b²-ab,b³-ab² ? Ab^2-ab,b^3-ab^2\in A, then b ? Ab\in A. We present a sufficient condition for the ring A[[X₁,?,X_n]]A[[X_1,\ldots ,X_n]] to be t-closed in B[[X₁,?,X_n]]B[[X_1,\ldots ,X_n]]. By an example, we show that our condition is not necessary. Even though the question is still open, some important cases are treated. For example, if A ì BA\subset B is an integral extension, or if A is p-injective, then A[[X₁,?,X_n]]A[[X_1,\ldots ,X_n]] is t-closed in B[[X₁,?,X_n]]B[[X_1,\ldots ,X_n]] if and only if A is t-closed in B.     

A remark on cuspidal local systems

In this note we show that all reductive groups are clean in characteristic ?3. In characteristic 2 there are two cuspidal local systems (one for F₄ and one for E₈) which can not be handled by our method.     

Principal quasi-Baerness of formal power series rings

Let R be a ring. We consider left （or right） principal quasi-Baerness of the left skew formal power series ring R[[x;α]] over R where a is a ring automorphism of R. We give a necessary and sufficient condition under which the ring R[[x; α]] is left （or right） principally quasi-Baer. As an application we show that R[[x]] is left principally quasi-Baer if and only if R is left principally quasi- Baer and the left annihilator of the left ideal generated by any countable family of idempotents in R is generated by an idempotent.     

Associative formal power series in two indeterminates

Investigating the associativity equation for formal power series in two variables we show that the transcendental associative formal power series are of order one or two and that they can be represented by an invertible formal power series in one variable. We also discuss the convergence of associative formal power series.     

Algebraic elements in formal power series rings

We obtain a theorem giving a condition for algebraicity of an element in a formal power series field of characteristicp>0. Using it many results can be proved, for example, the “theorem of the diagonal” of Furstenberg is deduced as an easy corollary. Dedicated to the memory of Chiyo Harase     

Two linear equations in formal power series

The equations [gradφ(x)]^TF(x)=h(x) and F(ψ(x))–ψ(x) are considered. They arise in the stability theory of differential and difference equations. The scalar function h(x) is a given, and the function ψ(x) an unknown, formal power series in the n indeterminates x=(x₁,…,x_n)^T, and h(0)=ψ=0; the elements of the n×n matrix F(x) are also formal power series in x, F(0)=0. It is shown that the solvability of both equations depends on the eigenvalues of the Jacobian F_x(0).     

Composition rings from formal power series rings

We study the ideals in some Dickson nearrings of polynomials and formal power series. For some of their related quotients, we introduce variants and generalizations, and construct composition rings also.     

Linear independence of automatic formal power series

We extend a result of J.-P. Allouche and O. Salon on linear independence of formal power series associated to polynomial extractions of quasistrongly p-additive sequences. The original result was on the F_p-linear independence and we extend it to the F_p[X]-linear independence.     

Commutator automorphisms of formal power series rings

For a big class of commutative rings , every continuous -automorphism of with the linear part the identity is in the commutator subgroup of . An explicit bound for the number of commutators involved and a -theoretic interpretation of this result are provided.