Some Pólya-Type Irreducibility Criteria for Multivariate Polynomials

We provide irreducibility criteria for multivariate polynomials with coefficients in an arbitrary field that extend a classical result of Pólya for polynomials with integer coefficients. In particular, we provide irreducibility conditions for polynomials of the form f(X)(Y ? f ₁(X))…(Y ? f _n (X)) + g(X), with f, f ₁, ?, f _n , g univariate polynomials over an arbitrary field.     

Automatic β-expansions of formal Laurent series over finite fields

We consider β-expansions of formal Laurent series over finite fields. If the base β is a Pisot or Salem series, we prove that the β-expansion of a Laurent series α is automatic if and only if α is algebraic.     

Irreducibility Criterion Over Finite Fields

The aim of this article is to prove the irreducibility of the polynomial Λ(Y) = Y ^d  + λ _d?1 Y ^d?1 + … + λ₀ over 𝔽 _q [X] where λ _i ∈ 𝔽 _q [X] and deg λ _d?1 > deg λ _i for each i ≠ d ? 1. We discuss in particular connections between the irreducible polynomials Λ and the number of Pisot elements in the case of formal power series.     

也谈实二次域类数的可除性

设ｄ无平方因子,ｈ（ｄ）是二次域Ｑ（ｄ）的类数,本文证明了：若１＋４ｋ２ｎ＝ｄａ２,ａ,ｋ＞１,ｎ＞２为正整数,且ａ＜０．９ｋ３５ｎ或ｎ的奇素因子ｐ和ｋ的素因子ｑ均适合（ｐ,ｑ－１）＝１,则除（ａ,ｄ,ｋ,ｎ）＝（５,４１,２,４）以外,ｈ（ｄ）≡０（ｍｏｄｎ）．同时,我们猜测：上述结果中的条件（ｐ,ｑ－１）＝１是不必要的．     

A residual transcendental extension of a valuation on K to K(x1, … , xn)

Let v be a valuation of a field K, G_v its value group and k_v its residue field. Let w be an extension of v to K(x₁, … , x_n). w is called a residual transcendental extension of v if k_w/k_v is a transcendental extension. In this study a residual transcendental extension w of v to K(x₁, … , x_n) such that transdegk_w/k_v = n is defined and some considerations related with this valuation are given.     

Independence of continued fractions in the field of Laurent series

Criteria for independence, both algebraic and linear, are derived for continued fraction expansions of elements in the field of Laurent series. These criteria are then applied to examples involving elements recently discovered to have explicit series and continued fraction expansions. Communicated by Attila Pethő     

Measurement of multiperiod aggregative efficiency

This article proposes a new method for measuring an aggregative efficiency of multiple period production systems. Every organization or firm generates a time series of data that represent its performances in the resource utilization and output production over multiple periods, and often desires an aggregated measure of efficiency for several periods of interest. Data envelopment analysis (DEA) has become an accepted and well-known approach to evaluating efficiency performance in a wide range of cases. However, most of the DEA studies have dealt primarily with ways to gauge the efficiency of production in only a single period so this efficiency reflects the insufficient or partial performance of multiple period productions. The new method is developed through extensions of the concept of Debreu–Farrell technical efficiency and is applied to evaluating the efficiency of cable TV service units with 3-year data.     

Multiple Laurent series and fundamental solutions of linear difference equations

Using the notion of fundamental solution, we obtain a solution to the Cauchy problem for a multidimensional homogeneous linear difference equation with constant coefficients.     

Levels of multi-continued fraction expansion of multi-formal Laurent series

The multi-continued fraction expansion of a multi-formal Laurent series is a sequence pair consisting of an index sequence and a multi-polynomial sequence . We denote the set of the different indices appearing infinitely many times in by H_∞, the set of the different indices appearing in by H₊, and call |H_∞| and |H₊| the first and second levels of , respectively. In this paper, it is shown how the dimension and basis of the linear space over F(z) (F) spanned by the components of are determined by H_∞ (H₊), and how the components are linearly dependent on the mentioned basis.     

Generalization of Stanley's monster reciprocity theorem

By studying the reciprocity property of linear Diophantine systems in light of Malcev-Neumann series, we present in this paper a new approach to and a generalization of Stanley's monster reciprocity theorem. A formula for the “error term” is given in the case when the system does not have the reciprocity property. We also give an inductive proof of Stanley's reciprocity theorem for linear homogeneous Diophantine systems.     

有一个实现包含Kr +1的可图序列

设K _{r +1}是一个r +1个顶点的完全图. 一个可图序列π =(d₁, d₂,…, d_n)称为是蕴含K _r+1 -可图的, 如果π有一个实现包含K _{r +1}作为子图. 该文进一步研究了蕴含K _{r+1 -}可图序列的一些新的条件, 证明了这些条件包含文献[14,10,11]中的一些主要结果和当n≥5r/2 +1时,σ(K _r+1, n)之值(此值在文献[2]中被猜测, 在文献[6,7,8,3]中被证实). 此外, 确定了所有满足n≥5, d₅≥4 且不蕴含K₅ -可图序列π=(d₁, d₂,…, d_n)的集合.     

On the value function in constrained control of Markov chains

It is known that the value function in an unconstrained Markov decision process with finitely many states and actions is a piecewise rational function in the discount factor a, and that the value function can be expressed as a Laurent series expansion about = 1 for close enough to 1. We show in this paper that this property also holds for the value function of Markov decision processes with additional constraints. More precisely, we show by a constructive proof that there are numbers O = _o <₁ <... < _m–1 < _m = 1 such that for everyj = 1, 2, ...,m – 1 either the problem is not feasible for all discount factors in the open interval (_j–1, _j) or the value function is a rational function in a in the closed interval [_j–1, _j]. As a consequence, if the constrained problem is feasible in the neighborhood of = 1, then the value function has a Laurent series expansion about = 1. Our proof technique for the constrained case provides also a new proof for the unconstrained case.     

罗朗级数环的主拟Baer性

称环 R为右主拟 Baer环（简称为右p·q.Baer环）,如果 R的任意主右理想的右零化子可由幂等元生成.本文证明了,若环 R满足条件Sl（R）（?）C（R）,则罗朗级数环R[[x,x-1]]是右p.q.Baer环当且仅当R是右p.q.Baer环且R的任意可数多个幂等元在I（R）中有广义join.同时还证明了,R是右p.q.Baer环当且仅当R[x,x-1]是右P.q.Baer环.