Field Of Laurent Series Research Articles

Logarithmic-exponential series

We extend the field of Laurent series over the reals in a canonical way to an ordered differential field of “logarithmic-exponential series” (LE-series), which is equipped with a well behaved exponentiation. We show that the LE-series with derivative 0 are exactly the real constants, and we invert operators to show that each LE-series has a formal integral. We give evidence for the conjecture that the field of LE-series is a universal domain for ordered differential algebra in Hardy fields. We define composition of LE-series and establish its basic properties, including the existence of compositional inverses. Various interesting subfields of the field of LE-series are also considered.

Commutative subalgebras of the ring of quantum polynomials and of the skew field of quantum Laurent series

A description of the commutative subalgebras of the ring of quantum polynomials and of the skew field of quantum Laurent series is given over a field of arbitrary characteristic.

Crystals via the affine Grassmannian

Let G be a connected reductive group over ℂ, and let $\mathfrak {g}$∨ be the Langlands dual Lie algebra. Crystals for $\mathfrak {g}$∨ are combinatorial objects that were introduced by M. Kashiwara (cf., e.g., [6]) as certain "combinatorial skeletons" of finite-dimensional representations of $\mathfrak {g}$∨. For every dominant weight λ of $\mathfrak {g}$∨ Kashiwara constructed a crystal B(λ) by considering the corresponding finite-dimensional representation of the quantum group Uq($\mathfrak {g}$∨) and then specializing it to q=0. Other (independent) constructions of B(λ) were given by G. Lusztig (cf. [9]) using the combinatorics of root systems and by P. Littelmann (cf. [7]) using the "Littelmann path model." It was also shown in [5] that the family of crystals B(λ) is unique if certain reasonable conditions are imposed (cf. Theorem 1.1). The purpose of this paper is to give another (rather simple) construction of the crystals B(λ) using the geometry of the affine Grassmannian \mathscr {G}$G=G($\mathscr{K}$)/G($\mathscr{O}$) of the group G, where $\mathscr{K}$=ℂ((t)) is the field of Laurent power series and $\mathscr{O}$=ℂ[[t]] is the ring of Taylor series. We then check that the family B(λ) satisfies the conditions of the uniqueness theorem from [5], which shows that our crystals coincide with those constructed in the references above. It would be interesting to find these isomorphisms directly (cf., however, [10]).

Automata, algebraicity and distribution of sequences of powers

Let K be a finite field of characteristic p. Let K((x)) be the field of formal Laurent series f(x) in x with coefficients in K. That is,

A Method for Solving the Hopf Equation and Generalization of Sobolev–Schwartz Distributions

We introduce distributions that are functionals with values in a nonarchimedian field of Laurent series. These objects naturally generalize Sobolev–Schwartz distributions, and we use them to find generalized solutions of the Hopf equation in the form of infinitely narrow solitons and shock waves. We propose a method for computation of the profile of an infinitely narrow soliton and of a shock wave described by the Hopf equation.

Irreducibility testing over local fields

The purpose of this paper is to describe a method to determine whether a bivariate polynomial with rational coefficients is irreducible when regarded as an element in Q((x))[y], the ring of polynomials with coefficients from the field of Laurent series in x with rational coefficients. This is achieved by computing certain associated Puiseux expansions, and as a result, a polynomial-time complexity bound for the number of bit operations required to perform this irreducibility test is computed.

Engel Expansions and the Rogers–Ramanujan Identities

We study the previously developed extension of the Engel expansion to the field of Formal Laurent series. We examine three separate aspects. First we consider the algorithm in relation to the work of Ramanujan. Second we show how the algorithm can be used to prove expansions such as those found by Euler, Rogers, and Ramanujan. Finally we remark briefly on its use in acceleration of convergence.

Diophantine Inequalities for Polynomial Rings

We study the Hardy–Littlewood method for the Laurent series field Fq((1/T)) over the finite field Fq with q elements. We show that if λ1, λ2, λ3 are non-zero elements in Fq((1/T)) satisfying λ1/λ2∉Fq(T) and sgn(λ1)+sgn(λ2)+sgn(λ3)=0,then the values of the sumλ1P1+λ2P2+λ3P3, as Pi (i=1, 2, 3) run independently through all monic irreducible polynomials in Fq[T], are everywhere dense on the “non-Archimedean” line Fq((1/T)), where sgn(f)∈Fq denotes the leading coefficient of f∈Fq((1/T)).

Fq-Linear Calculus over Function Fields

We define analogues of higher derivatives forFq-linear functions over the field of formal Laurent series with coefficients inFq. This results in a formula for Taylor coefficients of aFq-linear holomorphic function, a definition of classes ofFq-linear smooth functions which are characterized in terms of coefficients of their Fourier–Carlitz expansions. A Volkenborn-type integration theory forFq-linear functions is developed; in particular, an integral representation of the Carlitz logarithm is obtained.

Formal GNS Construction and States in Deformation Quantization

In this paper we develop a method of constructing Hilbert spaces and the representation of the formal algebra of quantum observables in deformation quantization which is an analog of the well-known GNS construction for complex C*-algebras: in this approach the corresponding positive linear functionals (“states”) take their values not in the field of complex numbers, but in (a suitable extension field of) the field of formal complex Laurent series in the formal parameter. By using the algebraic and topological properties of these fields we prove that this construction makes sense and show in physical examples that standard representations such as the Bargmann and Schrodinger representation come out correctly, both formally and in a suitable convergence scheme. For certain Hamiltonian functions (contained in the Gel'fand ideal of the positive functional) a formal solution to the time-dependent Schrodinger equation is shown to exist. Moreover, we show that for every Kahler manifold equipped with the Fedosov star product of Wick type all the classical delta functionals are positive and give rise to some formal Bargmann representation of the deformed algebra.

Approximation durch algebraische Zahlen beschr�nkten Grades im K�rper der formalen laurentreihen

Approximation by Algebraic Numbers of Bounded Degree in the Field of Formal Laurent Series. In this paper results ofWirsing andDavenport-Schmidt concerning the approximation of certain numbers by algebraic numbers of bounded degree are transferred to the field of formal power series connected with a certain non-archimedian valuation.

On Mahler's Classification in Laurent Series Fields

On Mahler's Classification in Laurent Series Fields

Division Algebras Not Embeddable in Crossed Products

We show that division algebras do not always embed in crossed product division algebras, so the latter do not serve as “Galois closures” for division algebras. We construct decomposable noncrossed product division algebras of prime-power index over rational function fields and Laurent series fields over number fields. We discover a rigid nonabelian group of order p 4.

ℚ(t) and ℚ((t))-Admissibility of Groups of Odd Order

Let _Q(t) be the rational function field over the rationals, Q, let Q((t)) be the Laurent series field over Q, and let W be a group of odd order. We investigate the following question: does there exist a finite-dimensional division algebra D central over Q(t) or Q((t)) which is a crossed product for W ? If such a D exists, W is said to be Q(t)-admissible (respectively, Q((t))-admissible). We prove that if W is Q((t))-admissible, then W is also Q(t)-admissible; we also exhibit a Q(t)-admissible group which is not Q((t))admissible. Let K be a field and let ' be a finite group. ' is said to be K-admissible if there exists a division algebra D, finite dimensional and central over K, which is a crossed product for '. Equivalently, ' is K-admissible if there exists a division algebra D with center K having a maximal subfield L Galois over K with Gal(L/K) S '. Admissibility questions for K = Q, the field of rational numbers, have been studied extensively in the literature (e.g., [Sc] and [ SO2 ]). More recently, results have been obtained when K is an algebraic function field over some field Ko ([FSS] and [FS]). In this paper we study admissibility questions for groups of odd order when K is either the rational function field Q(t) or the Laurent series field Q((t)). We show for such groups that Q((t))admissibility implies Q(t)-admissibility but not conversely. We also construct examples of groups of odd order which are Q((t))-admissible but which have homomorphic images which are not Q((t))-admissible; by contrast, if K is a number field, a homomorphic image of a K-admissible group is necessarily K-admissible [Sc, Corollary 2.3]. We fix below most of the basic terminology and notation that we will employ throughout this paper. Let K be a field. By a K-division algebra we mean a division algebra having center K which is finite dimensional over K. We say that A/K is central simple if A is a simple algebra with center K which is finite dimensional over K. Suppose A/K is central simple. By Wedderburn's Theorem, A _ M, (D) where D is a K-division algebra; we refer to D as the division algebra component of A. The Schur index of A, ind(A), equals Received by the editors September 21, 1993. 1991 Mathematics Subject Classification. Primary 12E1 5.

Sen’s theorem on iteration of power series

In the group of continuous automorphisms of the field of Laurent series in one variable over a field of characteristic p > 0 p > 0 , Sen’s Theorem describes the rapidity of convergence to the identity of the sequence formed by taking successive pth powers of a given element. This paper gives a short proof of Sen’s Theorem, utilizing the methods of p-adic analysis in characteristic zero.

𝑄(𝑡) and 𝑄((𝑡))-admissibility of groups of odd order

Let Q ( t ) \mathbb {Q}(t) be the rational function field over the rationals, Q \mathbb {Q} , let Q ( ( t ) ) \mathbb {Q}((t)) be the Laurent series field over Q \mathbb {Q} , and let G \mathcal {G} be a group of odd order. We investigate the following question: does there exist a finite-dimensional division algebra D central over Q ( t ) \mathbb {Q}(t) or Q ( ( t ) ) \mathbb {Q}((t)) which is a crossed product for G \mathcal {G} ? If such a D exists, G \mathcal {G} is said to be Q ( t ) \mathbb {Q}(t) -admissible (respectively, Q ( ( t ) ) \mathbb {Q}((t)) -admissible). We prove that if G \mathcal {G} is Q ( ( t ) ) \mathbb {Q}((t)) -admissible, then G \mathcal {G} is also Q ( t ) \mathbb {Q}(t) -admissible; we also exhibit a Q ( t ) \mathbb {Q}(t) -admissible group which is not Q ( ( t ) ) \mathbb {Q}((t)) -admissible.

On the distribution of 𝑘-dimensional vectors for simple and combined Tausworthe sequences

The lattice structure of conventional linear congruential random number generators (LCGs), over integers, is well known. In this paper, we study LCGs in the field of formal Laurent series, with coefficients in the Galois field F 2 {\mathbb {F}_2} . The state of the generator (a Laurent series) evolves according to a linear recursion and can be mapped to a number between 0 and 1, producing what we call a LS2 sequence. In particular, the sequences produced by simple or combined Tausworthe generators are special cases of LS2 sequences. By analyzing the lattice structure of the LCG, we obtain a precise description of how all the k-dimensional vectors formed by successive values in the LS2 sequence are distributed in the unit hypercube. More specifically, for any partition of the k-dimensional hypercube into 2 k l {2^{kl}} identical subcubes, we can quickly compute a table giving the exact number of subcubes that contain exactly n points, for each integer n. We give numerical examples and discuss the practical implications of our results.

Dérivations, et hautes dérivations, dans certains corps gauches de series de Laurent

Let k be a field of characteristic zero, F=k((Y)) the local field of Laurent series in a variable Y, and ∂ Y the usual derivation of F. The purpose of this paper is the complete description of k-derivations on the formal pseudo-differential operator skewfield K=F(XX, ∂ Y )). An application of this result is given to study the structure of higher derivations in skewfields of characteristic zero

Some explicit continued fraction expansions

We determine infinite products in the field of Laurent series with the property that the truncations of the product yield every second continued fraction convergent of the product. We mention some related examples and specialize to obtain numerical results.

The number of convolutional codes of span one

A convolutional code is a space over the field of Laurent series over a finite field which is generated by a polynomial matrix. In this paper, we count the number of convolutional codes which are generated by polynomial matrices of degree one.