Power Series In Several Variables Research Articles

Riordan Groups in Higher Dimensions

Abstract: The classical Riordan groups associated to a given commutative ring are groups of infinite matrices (called Riordan arrays) associated to pairs of formal power series in one variable. The Fundamental Theorem of Riordan Arrays relates matrix multiplication to two group actions on such series, namely formal (convolution) multiplication and formal composition. We define the analogous Riordan groups involving formal power series in several variables, and establish the analogue of the Fundamental Theorem in that context. We discuss related groups of Laurent series and pose some questions.

Semigroup associated with a free polynomial

Let K be an algebraically closed field of characteristic zero and let KC〚x1,⋯,xe〛 be the ring of formal power series in several variables with exponents in a line free cone C. We consider irreducible polynomials f=yn+a1(x_)yn−1+⋯+an(x_) in KC〚x1,⋯,xe〛[y] whose roots are in KC〚x11n,⋯,xe1n〛. We generalize to these polynomials the theory of Abhyankar-Moh. In particular we associate with any such polynomial its set of characteristic exponents and its semigroup of values. We also prove that the set of values can be obtained using the set of approximate roots. We finally prove that polynomials of K〚x1,⋯,xe〛[y] fit in the above set for a specific line free cone (see Section 4).

On the Asymptotic Expansion of Monomial in Singular Partial Differential Equations

People find that formal solutions to a type of doubly singular ordinary differential equations (systems) summable in a monomial of two variables, and that formal solutions to many singular partial differential equations were multisummable. So summability theory of formal power series in several variables is of great importance to the research of formal solutions to partial differential equations. This enriches the achievements in the research of formal solutions to differential equations. It is a application of monomial summability theory of formal power series.

Power series of several variables with condition of logarithmical convexity

We obtain a new version of Hardy theorem about power series of several variables reciprocal to the power series with positive coefficients. We prove that if the sequence {as} = as1,s2,...,sn, ||s|| ≥ K satisfies condition of logarithmically convexity and the first coefficient a0 is sufficiently large then reciprocal power series has only negative coefficients {bs} = bs1,s2,...,sn, except b0,0,...,0 for any K. The classical Hardy theorem corresponds to the case K = 0, n = 1. Such results are useful in Nevanlinna - Pick theory. For example, if function k(x, y) can be represented as power series Σn≥0 an(x-y)n, an > 0, and reciprocal function 1/k(x,y) can be represented as power series Σn≥0 bn(x-y)n such that bn < 0, n > 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna-Pick property. The reproducing kernel 1/1-x-y of the classical Hardy space H2(D) is a prime example for our theorems.

Power Series of Several Variables with the Condition of the Logarithmical Convexity of the Coefficients

In the paper, the generalization of the Hardy theorem for power series of several variables inverse to the power series with positive coefficients is considered. It is proved that if the sequence of coefficients {as} = $${{a}_{{{{s}_{1}},{{s}_{2}}, \ldots ,{{s}_{n}}}}}$$ of a power series satisfies the condition similar to the condition of the logarithmical convexity of the coefficients and the first coefficient a0 is sufficiently large starting from a certain place ||s|| ≥ K, then all the coefficients of the inverse power series are negative, except the first coefficient. The classical Hardy theorem corresponds to the case K = 0 and n = 1. These results are applicable in the Nevanlinna–Pick theory.

Formal power series in several variables

Formal power series in several variables

Bohr Inequalities on Noncommutative Polydomains

The goal of this paper is to study the Bohr phenomenon in the setting of free holomorphic functions on noncommutative regular polydomains $$\mathbf{D_f^m}$$ , $$\mathbf{f}=(f_1,\ldots , f_k)$$ , generated by positive regular free holomorphic functions. These polydomains are noncommutative analogues of the scalar polydomains $$\begin{aligned} {{\mathcal {D}}}_{f_1} ({{\mathbb {C}}})\times \cdots \times {{\mathcal {D}}}_{f_k}({{\mathbb {C}}}), \end{aligned}$$ where each $${{\mathcal {D}}}_{f_i}({{\mathbb {C}}})\subset {{\mathbb {C}}}^{n_i}$$ is a certain Reinhardt domain generated by $$f_i$$ . We characterize the free holomorphic functions on $$\mathbf{D_f^m}$$ in terms of the universal model of the polydomain and extend several classical results from complex analysis to our noncommutative setting. It is shown that the free holomorphic functions admit multi-homogeneous and homogeneous expansions as power series in several variables. With respect to these expansions, we introduce the Bohr radii $$K_{mh}(\mathbf{D_f^m})$$ and $$K_{h}(\mathbf{D_f^m})$$ for the noncommutative Hardy space $$H^\infty (\mathbf{D_{f,\text { rad}}^m})$$ of all bounded free holomorphic functions on the radial part of $$\mathbf{D_f^m}$$ . Several well-known results concerning the Bohr radius associated with classes of bounded holomorphic functions are extended to our noncommutative multivariable setting.

About the algebraic closure of the field of power series in several variables in characteristic zero

We begin this paper by constructing different algebraically closed fields containing an algebraic closure of the field of power series in several variables over a characteristic zero field. Each of these fields depends on the choice of an Abhyankar valuation and is constructed via a generalization of the Newton-Puiseux method for this valuation. Then we study the Galois group of a polynomial with power series coefficients. In particular by examining more carefully the case of monomial valuations we are able to give several results concerning the Galois group of a polynomial whose discriminant is a weighted homogeneous polynomial times a unit. One of our main results is a generalization of Abhyankar-Jung Theorem for such polynomials, classical Abhyankar-Jung Theorem being devoted to polynomials whose discriminant is a monomial times a unit.

Spherically balanced Hilbert spaces of formal power series in several variables

Spherically balanced Hilbert spaces of formal power series in several variables

Matrix Factorizations in Algebra, Geometry, and Physics

Let $W$ be a polynomial or power series in several variables, or, more generally, a nonzero element in some regular commutative ring. A matrix factorization of $W$ consists of a pair of square matrices $X$ and $Y$ of the same size, with entries in the given ring, such that the matrix product $XY$ is $W$ multiplied by the identity matrix. For example, if $X$ is a matrix whose determinant is $W$ and $Y$ is its adjoint matrix, then $(X, Y)$ is a matrix factorization of $W$. Such matrix factorizations are nowadays ubiquitous in several different fields in physics and mathematics, including String Theory, Commutative Algebra, Algebraic Geometry, both in its classical and its noncommutative version, Singularity Theory, Representation Theory, Topology, there in particular in Knot Theory. The workshop has brought together leading researchers and young colleagues from the various input fields; it was the first workshop on this topic in Oberwolfach. For some leading researchers from neighboring fields, this was their first visit to Oberwolfach.

Tropical analytic geometry, Newton polygons, and tropical intersections

In this paper we use the connections between tropical algebraic geometry and rigid-analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in several variables: if f1,…,fn are n convergent power series in n variables with coefficients in a non-Archimedean field K, we give a formula for the valuations and multiplicities of the common zeros of f1,…,fn. We use rigid-analytic methods to show that stable complete intersections of tropical hypersurfaces compute algebraic multiplicities even when the intersection is not tropically proper. These results are naturally formulated and proved using the theory of tropicalizations of rigid-analytic spaces, as introduced by Einsiedler, Kapranov, and Lind (2006) [14] and Gubler (2007) [20]. We have written this paper to be as readable as possible both to tropical and arithmetic geometers.

Polynomial Imaginary Decompositions for Finite Separable Extensions

Let K be a field and let L = K[ξ] be a finite field extension of K of degree m > 1. If f ∈ L[Z] is a polynomial, then there exist unique polynomials u0, . . . , um−1 ∈ K[X0, . . . , Xm−1] such that f( Pm−1 j=0 ξ Xj) = Pm−1 j=0 ξ uj . A. Nowicki and S. Spodzieja proved that, if K is a field of characteristic zero and f 6= 0, then u0, . . . , um−1 have no common divisor in K[X0, . . . , Xm−1] of positive degree. We extend this result to the case when L is a separable extension of a field K of arbitrary characteristic. We also show that the same is true for a formal power series in several variables.

Sur la linéarité de la fonction de Artin

We give here a counter-example to an old conjecture in the theory of singularities. This conjecture is that the function that appears in the strong Artin approximation theorem is bounded by an affine function. First we study Diophantine approximation between the field of power series in several variables and its completion for the m -adic topology. We show, with an example, that there is no Liouville theorem in this case. This example gives us our counter-example (cf. théorème 1.2). As an application, we give a new proof of the fact that there is no theory of elimination of quantifiers for the field of fractions of the ring of power series in several variables.

Summability of power series in several variables, with applications to singular perturbation problems and partial differential equations

Nous introduisons et considerons une methode de sommabilite de series de puissances en plusieures variables et. nous donnons des applications aux solutions formelles de problemes de perturbations singulieres et aux equations aux derivees partielles. Nous etendons ainsi des resultats de Lutz, Miyake et Schaefke, resp. Balser, pour l'equation de la chaleur variables complexes pour (les cas plus generaux.

Bounds for quotients in rings of formal power series with growth constraints

In rings $ {\mit \Gamma }_M $ of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence $ M=(M_l )_{ l \geq 0} $ (such as rings of Gevrey series), we find precise estimates for quotients $ F/{\mit \Phi }

Rigidity Theorems for Multiparametric Deformations of the Universal Enveloping Algebras

The well-known one-dimensional Riemann–Hilbert problem on C concerns the proof of the existence of the Fuchsian system on C with a given monodromy, that is, with a given representation of a fundamental group, which is a braid group. In higher dimensions, the classical statement of the problem is modified so that instead of proving the existence of a system of differential equations of Fuchsian type, the problem of characterization of the representations of a fundamental group that can be a monodromy representation of the differential system of Fuchsian type is stated. It is a purely algebraic problem, but since it is closely connected with the question concerning the recovery of the functions of several complex variables if their ramification is given, its solution is based on analytical methods connected with properties of the power series on the parameters of the differential equation. In the context of the Riemann–Hilbert problem, to these methods should be added some algebraic ones concerning the algebraic structures that can be defined on algebras of such series. Some of these methods are considered in this paper, in particular, the method of multi-parametric deformations of some algebras, more exactly, the quantum enveloping algebras. In the one-parametric case, the quantum enveloping algebras are the algebras over the ring of the formal power series C[[h]]. The last object carries the topological structure defined by the powers of an ideal generated by a (quantum) parameter h. This topological structure naturally induces some structures on algebras and on modules over algebras. All methods of proving the different statements about these structures naturally are based on induction on the powers of h. Obviously, such a technique can be extended to the formal power series of several variables C[[h1, . . . , hn]]. Since many of the proofs of the theorems are easily carried over from the one-parametric case to the multi-parametric one, we have hopes that a variety of facts from the classical theory of quantum groups are valid for multi-parametric deformations. This paper is devoted to problems connected with multi-parametric generalizations of methods that are important for the multi-parametric analogue of the Drinfeld–Kohno theory. The well-known Drinfeld–Kohno theorem is a statement about the conditions of solvability of the Riemann–Hilbert problem in a class of the Knizhnik–Zamolodchikov equations [2, 7], which are particular cases of the multidimensional Fuchsian systems. Their proof and some of their generalizations can be found in [2,3,5,7]. In this paper, we first introduce some definitions that are used in the extensions of the Drinfeld– Kohno theory: the definition of the multi-parametric Drinfeld–Jimbo algebra and definitions of the braided multi-parametric quantum enveloping algebra and the braided multi-parametric quasi-bialgebra. We introduce the isomorphism of the quasi-bialgebras up to the twisting. Further, the uniqueness theorems for the homomorphisms of the trivial multi-parametric deformation of the enveloping algebra if these homomorphisms coincide on the universal enveloping algebras are proved. Then we prove some theorems on the uniqueness for the quantum enveloping algebras and for the Drinfeld–Jimbo algebra. We also prove the theorem on the isomorphism (up to the twisting) of the braided quasi-bialgebras having different associators. All these results are the necessary components in a proof of the multi-parametric analogue of the Drinfeld theorem on the isomorphism of the structure of a braided quasi-bialgebra (on

A Certain Formal Power Series of Several Variables Attached to Local Densities of Quadratic Forms, I

We study the power series attached to local densities defined by the author ( Tôholu Math. J. 44 (1992), 211-218). In particular, using a certain induction formula for local densities, we will determine an explicit form of its denominator.

Embedding the Algebra of Formal Power Series in Several Variables into a Banach Algebra

Embedding the Algebra of Formal Power Series in Several Variables into a Banach Algebra

Laminar three-dimensional mixed convection about a rotating sphere in a stream

The power series of several variables has been applied to study the laminar mixed convection about an isothermal rotating sphere in a stream of arbitrary direction with respect to the axis of rotation, so that the velocity profile is three-dimensional. The boundary layer equations are numerically solved and results are presented for values of the rotation parameter and the buoyancy parameter ranging from 0 to 10. For some particular cases, qualitative and quantitative comparisons with previous works reported in the literature agree with each other. Moreover, the agreement between the theoretical results and the experimental data is satisfactory.

Algebraic power series and diagonals

We extend some results of Christol and Furstenberg to the case of several variables: (1) A power series in several variables over the p-adic integers Z p is congruent mod p s to an algebraic power series if and only if its coefficients satisfy certain congruences mod p s . (2) Any algebraic power series in m variables over a field K can be written as the diagonal of a rational power series in 2 m variables. We also give an elementary proof of a result of Deligne: The diagonal of an algebraic power series in several variables, over a field of nonzero characteristic is algebraic. Moreover we show that the diagonal of an algebraic power series over Z p is algebraic mod p s , for every s. We also obtain other related results.