The digit generating function of a polynomial

The average frequency of 1 occurring as the kth digit in the binary expansion of squares, cubes, and generally the values of a polynomial is studied. In particular, it turns out that the generating function of these frequencies is rational for the important special cases of powers, linear and quadratic polynomials. For higher degree polynomials, the behaviour seems to be much more chaotic in general, which is exhibited by two examples of cubic polynomials.     

A note on the interlacing of zeros and orthogonality

Let be a sequence of monic polynomials with deg(t_n)=n such that, for each n∈N, the zeros of t_n are real and simple and t_n and t_n+1 have no common zeros. We discuss the connection between the orthogonality of the sequence, the positivity of a certain ratio, and the interlacing of the zeros of t_n and t_n+1 for n≥1, n∈N.     

Polynomials Contained in a Finite Number of Maximal Ideals

Abstract

Let R[X] be a polynomial ring in one variable over a commutative ring R. If (R,?) is a local ring then any Weierstrass polynomial in R[X] is contained only in the maximal ideal (?,X) of R[X]. We generalise this property of Weierstrass polynomials and investigate properties of polynomials contained in a finite number of maximal ideals in R[X].     

关于积分上限函数的几个定理

变上限积分是积分学的一个重要理论，其运算结果仍以函数的形式体现．研究这类函数，得出几个颇有理论意义的定理。     

环中关于理想同余的粗糙集的性质

讨论了环中由理想同余关系所决定的下、上近似的一些特殊性质．     

Primary Ideals and Valuation Ideals in Semigroups

Let S be a cancellation torsion-free additive semigroup with identity 0 and let S {0}. We consider the relation between the valuation ideals and primary ideals of S. We give a characterization that each primary ideal is a valuation ideal in S and also give a characterization that each valuation ideal is a primary ideal in S.AMS Subject Classification (1991): 20M14     

A Remark on a Theorem of Szegö

Let be a polynomial with complex coefficients and define, for , where ||P|| is the euclidean norm of the polynomial P. By a theorem of Szegö where is the Mahler measure of F. Recently, J. Dégot proved an effective version of this result. In this paper we sharpen Dégot's result, under the additional hypotheses that F is a square-free polynomial with integer coefficients and without reciprocal factors.     

A Relative Filtration Index and Fibers of Normal Primes in Extensions of Finite Type

For an extension R A of commutative Noetherian rings the behavior of the associated morphism of topological spaces Spec A Spec R is often measured by its behavior on each of its fibers. Specifically, one studies the 'splitting' (or 'branching') and the 'ramification' that occurs in each fiber. In the classical constructions of faithfully flat analytic extensions (e.g., completion or Henselization) of excellent local rings the splitting and ramification properties are fairly well understood; see EGA IV [6, 18.10], Nagata [13, Sect. 37] or Raynaud [15, Ch. IX]. The strongest results are usually achieved for fibers over a 'normal point' of Spec R, that is, over p Spec R such that R/p is a normal domain [e.g., the property of a normal prime p in a local ring to be 'unibranched', i.e., the Henselization of R/p is a (normal) domain].     

探讨Abel积分零点的一个定理

本文研究了具有中心环域的可积多项式系统,探讨此系统在用多项式进行微扰的情况下,其所对应的Abel积分的零点个数的计算方法,给出了一个较为实用的定理,并举出了若干个应用的例子．     

On Primitive Ideals in Polynomial Rings over Nil Rings

Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indeterminate over R are of the form I[x] for some ideals I of R. Presented by S. MontgomeryMathematics Subject Classifications (2000) 16N40, 16N20, 16N60, 16D25.Agata Smoktunowicz: Current address: Institute of Mathematics, Polish Academy of Sciences, 00-956 Warsaw 10, niadeckich 8, P.O. Box 21, Poland.     

整系数多项式因式分解的一种新方法

利用整系数多项式与正有理数的对应 ,将多项式因式分解通过对真分数序列筛选的办法求得因式 ,给出了整系数多项式因式分解的一种新方法 .     

THE HENSTOCK INTEGRAL FOR FUNCTIONS WITH VALUES IN NUCLEAR SPACES AND THE HENSTOCK LEMMA

This paper shows that Henstock‘s Lemma holds for functions with values in a countably Hilbert space, where the Henstock integral is defined as a natural extension of the resl valued case.