Irreducibility of A Formal Power Series with Integer Coefficients

Two algorithms are introduced and shown to lead to a unique product represenatation for a given formal power series 4 with constant term 1. as a product of terms where the a n are polynomials. The degree of approximation by the rational function which is the Nth partial product of such terms for n≤N is investigated. The type of product corresponding to an initial rational function A is also characterized, as well as the product in some other explicit cases.

Mahler functions are power series f(x) with complex coefficients for which there exist a natural number n and an integer l ≥ 2 such that f(x), f(xl),..., $$f({x^{{\ell ^{n - 1}}}}),f({x^{{\ell ^n}}})$$ are linearly dependent over ℂ(x). The study of the transcendence of their values at algebraic points was initiated by Mahler around the’ 30s and then developed by many authors. This paper is concerned with some arithmetic aspects of these functions. In particular, if f(x) satisfies f(x) = p(x)f(xl) with p(x) a polynomial with integer coefficients, we show how the behaviour of f(x) mirrors on the polynomial p(x). We also prove some general results on Mahler functions in analogy with G-functions and E-functions.

Chapter 2 - Series

Let \({\mathcal{O}}\) be the local ring at a singular point of a geometrically integral algebraic curve defined over a finite field, and let m be the number of branches centered at the curve singularity. By encoding cardinalities of certain finite sets of ideals, we associate to each pair of ideal classes of \({\mathcal{O}}\) a power series in m variables with integer coefficients, which can be represented by an integral within the framework of harmonic analysis. We prove that partial local zeta functions can be expressed in terms of these multi-variable power series. The main objective of this paper is to investigate the properties of these series, and to provide in this way a deeper insight into the nature of local zeta functions.

Formal power series and regular operations on fuzzy languages

In this chapter we prepare the ground for the proof of the Jung-Abhyankar theorem in section 2 and the study of quasiordinary power series in section 4 of chapter V. We assume that the reader is acquainted with the notion of power series over a field; in section 1, for the convenience of the reader, we give some background, introduce convergent power series over the real and complex numbers in section 2, and prove Weierstraß division and preparation theorem in section 3. The category of formal (resp. analytic) algebras—these are homomorphic images of formal (resp. convergent) rings of power series—is treated in section 4. The last section considers, in particular, the completion of an analytic algebra; these results are needed to prove the convergent case of the Jung-Abhyankar theorem.

Reduction of everywhere convergent power series with respect to Gröbner bases

In this chapter, we consider a system of differential equations $$x\frac{{d\vec{y}}}{{dx}} = \vec{f}\left( {x,\vec{y}} \right)$$ (E) assuming that the entries of the \({{\mathbb{C}}^{n}}\)-valued function \(\vec{f}\) are convergent power series in complex variables \(\left( {x,\vec{y}} \right) \in {{\mathbb{C}}^{{n + 1}}}\) with coefficients in ℂ, where x is a complex independent variable and \(\vec{y} \in {{\mathbb{C}}^{n}}\) is an unknown quantity. The main tool is calculation with power series in x. In §I-4, using successive approximations, we constructed power series solutions. However, generally speaking, in order to construct a power series solution \(\vec{y}\left( x \right) = \sum\limits_{{m = 1}}^{\infty } {{{x}^{m}}{{{\vec{a}}}_{m}}}\) this expression is inserted into the given differential equation to find relationships among the coefficients \({{\vec{a}}_{m}}\), and the coefficients \({{\vec{a}}_{m}}\) are calculated by using these relationships. In this stage of the calculation, we do not pay any attention to the convergence of the series. This process leads us to the concept of formal power series solutions (cf. §V-1). Having found a formal power series solution, we estimate \(|{{\vec{a}}_{m}}|\) to test its convergence. As the function \({{x}^{{ - 1}}}\vec{f}\left( {x,\vec{y}} \right)\) is not analytic at x = 0, Theorem I-4-1 does not apply to system (E). Furthermore, the existence of formal power series solutions of (E) is not always guaranteed. Nevertheless, it is known that if a formal power series solution of (E) exists, then the series is always convergent. This basic result is explained in §V-2 (cf. [CL, Theorem 3.1, pp. 117-119] and [Wasl, Theorem 5.3, pp. 22-25] for the case of linear differential equations]). In §V-3, we define the S-N decomposition for a lower block-triangular matrix of infinite order. Using such a matrix, we can represent a linear differential operator $$ \mathcal{L}\left[ {\vec y} \right]{\text{ = }}x\frac{{d\vec y}}{{dx}}{\text{ + }}\Omega {\text{(}}x{\text{)}} \vec y, $$ (LDO) Where \(\Omega \left( x \right)\) is an n x n matrix whose entries are formal power series in x with coefficients in ℂ n . In this way, we derive the S-N decomposition of L in §V-4 and a normal form of L in §V-5 (cf. [HKS]). The S-N decomposition of L was originally defined in [GerL]. In §V-6, we calculate the normal form of a given operator L by using a method due to M. Hukuhara (cf. [Si17, §3.9, pp. 85-89]). We explain the classification of singularities of homogeneous linear differential equations in §V-7. Some basic results concerning linear differential equations given in this chapter are also found in [CL, Chapter 4].

A power series is called lacunary if “almost all” of its coefficients are zero. Integer partitions have motivated the classification of lacunary specializations of Han’s extension of the Nekrasov–Okounkov formula. More precisely, we consider the modular forms Fa,b,c(z):=η(24az)aη(24acz)b-aη(24z),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned}F_{a,b,c}(z) :=\\frac{\\eta (24az)^a \\eta (24acz)^{b-a}}{\\eta (24z)},\\end{aligned}$$\\end{document}defined in terms of the Dedekind eta -function, for integers a,c ge 1, where b ge 1 is odd throughout. Serre (Publications Mathématiques de l’IHÉS 123–201:2959–2968, 1981) determined the lacunarity of the series when a = c = 1. Later, Clader et al. (Am Math Soc 137(9):2959–2968, 2009) extended this result by allowing a to be general and completely classified the F_{a,b,1}(z) which are lacunary. Here, we consider all c and show that for {a in {1,2,3}}, there are infinite families of lacunary series. However, for a ge 4, we show that there are finitely many triples (a, b, c) such that F_{a,b,c}(z) is lacunary. In particular, if a ge 4, b ge 7, and c ge 2, then F_{a,b,c}(z) is not lacunary. Underlying this result is the proof the t-core partition conjecture proved by Granville and Ono (Trans Am Math Soc 348(1):331–347, 1996).

We consider power series over a graded monoid \(M\) of finite type. We show first that, under certain conditions, the equivalence problem of power series over \(M\) with coefficients in the semiring \(\mathbb N\) of nonnegative integers can be reduced to the equivalence problem of power series over \(\{x\}^*\) with coefficients in \(\mathbb N\). This result is then applied to rational and recognizable power series over \(M\) with coefficients in \(\mathbb N\), and to rational power series over \(\Sigma ^*\) with coefficients in the semiring \(\mathbb {Q}_+\) of nonnegative rational numbers, where \(\Sigma \) is an alphabet.

Coset Diagrams for a Homomorphic Image of Δ (3, 3, k)

We study the Hurwitz product (convolution) in the space of formal Laurent series over an arbitrary field of zero characteristic. We obtain the convolution equation which is satisfied by the Euler series. We find the convolution representation for an arbitrary differential operator of infinite order in the space of formal Laurent series and describe translation invariant operators in this space. Using the p-adic topology in the ring of integers, we show that any differential operator of infinite order with integer coefficients is well defined as an operator from ℤ[[z]] to ℤp[[z]].

The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r+1 elements but no fewer. These results are implemented in the Stafford package for D=A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series.

Central and local limit theorems for the coefficients of polynomials of binomial type

We study certain properties of the group of substitutions of formal power series in one variable with integer coefficients. We show that , regarded as a topological group, has four generators and cannot be generated by fewer elements. In particular, we show that the one-dimensional continuous homology of is isomorphic to . We study various topological and geometric properties of the coset space . We compute the real cohomology with uniformly locally constant supports and show that it is naturally isomorphic to the cohomology of the nilpotent part of the Lie algebra of formal vector fields on the line.