Z/pkZ上的多元置换多项式

设Z表整数集,p为一给定奇素数,k为一正整数.张起帆(1995)得到了一类模p奇异的二元多项式成为剩余类环Z/pkZ上的置换多项式的一个充要条件,胡永忠(2001)将张起帆的这一结果的充分条件推广到了一般n元的情形.本文得到了一类模p奇异的n元多项式成为剩余类环Z/pkZ上的置换多项式的一个充要条件,所得结论是对张文(n=2)和胡文(n＞2)的自然推广和改进.     

关于高斯整数环的商环元素个数的注记

设 Z表示整数环 ,i表示虚数单位 ( i=- 1 ) .Z( i)为所有形如 a+ bi( a,b∈ Z)的复数组成的集合 ,称为高斯整数环 .高斯整数环中的元素称为高斯整数 .在文 [1 ]中 ,提出了两个猜测 ,其中之一是 :设 m和 n都是整数 ,则高斯整数环 Z( i)的商环 Z( i) /( m+ ni)的元素个数不超过 m2 + n2 .本文证明这一结论成立 ,且更明确的有 ,| Z( i) /( m+ ni) | =m2 + n2 .注意 ,对 m=0 (或 n=0 )以及 m任意但 n=1 (或 n任意但 m=1 )的情形 ,文 [1 ]已经证明此等式成立 .以下我们用 | A|表示集合 A的元素个数 ,也用 | α|表示复数 α的模 .下面给出的是…     

关于不定方程x~2+4~n=y~(13)(n=4,5,6)的整数解

在高斯整环中,利用代数数论与同余理论的方法,讨论了不定方程x~2+4~n=y~(13)(n=4,5,6)的整数解问题,得出了当n=4,5时无整数解;n=6是仅有整数解(x,y)=(64,2)和(x,y)=(-64,2)的结论,推进了不定方程整数解的研究.     

Z_n[i]的平方映射图（英文）

本文研究了模n高斯整数环Z_n[i]的平方映射图Γ(n).利用数论、图论与群论等方法,获得了Γ(n)中顶点0及1的入度,并研究了Γ(n)的零因子子图的半正则性.同时,获得了Γ(n)中顶点的高度公式.推广了Somer等人给出的模n剩余类环平方映射图的相关结论.     

整系数多项式的哥德巴赫定理

哥德巴赫猜想断定每一个比4大的偶数是两个素数的和、用Z表示整数环,多项式环Z[x]与Z一样是一个唯一分解整环,其中不可约多项式相当于整数中的素数。本文的目的是证明多项式环Z[x]中与哥德巴赫猜想类似的定理。 定理1.在Z[x]中每一个次数n≥1的多项式M可以写成两个不可约n次多项式A与B的和,即     

关于n连贯多项式的几个结论

数集K上的多项式f（x）＋i（i＝0，1，…，n－1，整数n≥2）均在K上可约，则称f（x）为K上的n连贯多项式，二连贯多项式简称连贯多项式．自[1]提出n连贯多项式的概念以来，有较多文献在研究它．一般在复数集C，实数集R，有理数集Q，或整数集Z上研究n连贯多项式．本交给出关于”连贯多项式的n个结论（没有指明在哪个数集上时，指在任意数集上），这些结论都是由n连贯多项式的定义容易证明的，所以多未证明．定理1（1）复数域C上次数不小于2，或R上次数不小于3的多项式均为n连贯多项式；（2）ax2＋bx＋c（a＞0）在R上为n连贯多项式的充要…     

论一个由递推关系给出的多项式序列

一、引言笔者曾在一个存在性问题的研究中,偶然地引出了如下一个由递推关系给出的多项式序列{f_n(x)}: f_o(x)=1,f_1(x)=r, f_n(x)=xf_(n-1)(x)-f_(n-2)(x),(n≥2)(1) 尽管其存在性问题早已解决,但由此多项式序列又意外地得到了几个有趣的组合恒等式以及一系列三角恒等式,同时还发现了一类三角函数式的求值方法。故书拙文,以求同行斧正, 二、f_n(x)的表达式与f_n(x)的根由于f_n(x)是x的多项式,因而自然地想求出它的表达式,容易用数学归纳法证明下面的定理1 对任意非负整数n,有其中[t]表示不超过实数t的最大整数。(证略) 当n≥5时,n次多项式的根无公式解,因     

两类递归图的Tutte多项式

本文用转移矩阵给出了C2×Pn和N2∨Pn的Tutte多项式的公式,根据公式设计了M ap le程序,该程序对每个固定的整数n 2都能计算出C2×Pn和N2∨Pn的Tutte多项式.     

第34届IMO试题解答

1 设f(x)=x~n+5x~(n-1)+3,其中n是一个大于1的整数。求证:f(x)不能表示为两个多项式的乘积,其中每一个多项式都具有整数系数而且它们的次数都不低于一次。解假设f(x)可分解为两个整系数多项式之积     

联合最佳有理逼近

在实连续函数空间C[a,b]中定义一致范数。记(?)_N为次数≤N的所有代数多项式所成之集合,其中N为非负整数。记m,n为固定的非负整数,令(为简单计,Q>0(≥0)表示Q(x)>0(≥0),(?)x∈[a,b])     

Monic integer Chebyshev problem

We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let denote the monic polynomials of degree with integer coefficients. A monic integer Chebyshev polynomial satisfies

and the monic integer Chebyshev constant is then defined by

This is the obvious analogue of the more usual integer Chebyshev constant that has been much studied.

We compute for various sets, including all finite sets of rationals, and make the following conjecture, which we prove in many cases.

Conjecture. Suppose is an interval whose endpoints are consecutive Farey fractions. This is characterized by Then

This should be contrasted with the nonmonic integer Chebyshev constant case, where the only intervals for which the constant is exactly computed are intervals of length 4 or greater.

     

Parametric methods in integer linear programming

In contrast to methods of parametric linear programming which were developed soon after the invention of the simplex algorithm and are easily included as an extension of that method, techniques for parametric analysis on integer programs are not well known and require considerable effort to append them to an integer programming solution algorithm.The paper reviews some of the theory employed in parametric integer programming, then discusses algorithmic work in this area over the last 15 years when integer programs are solved by different methods. A summary of applications is included and the article concludes that parametric integer programming is a valuable tool of analysis awaiting further popularization.     

Analysis of PSLQ, an integer relation finding algorithm

Let be either the real, complex, or quaternion number system and let be the corresponding integers. Let be a vector in . The vector has an integer relation if there exists a vector , , such that . In this paper we define the parameterized integer relation construction algorithm PSLQ, where the parameter can be freely chosen in a certain interval. Beginning with an arbitrary vector , iterations of PSLQ will produce lower bounds on the norm of any possible relation for . Thus PSLQ can be used to prove that there are no relations for of norm less than a given size. Let be the smallest norm of any relation for . For the real and complex case and each fixed parameter in a certain interval, we prove that PSLQ constructs a relation in less than iterations.

     

Corrigendum: Combinatorial properties of integer matrices and integer matrices mod k

We correct an error in our paper “Combinatorial Properties of Integer Matrices and Integer Matrices mod k” that appeared in this journal (66, 1380–1402 (2017)).     

An integer linear programming approach for a class of bilinear integer programs

We propose an Integer Linear Programming (ILP) approach for solving integer programs with bilinear objectives and linear constraints. Our approach is based on finding upper and lower bounds for the integer ensembles in the bilinear objective function, and using the bounds to obtain a tight ILP reformulation of the original problem, which can then be solved efficiently. Numerical experiments suggest that the proposed approach outperforms a latest iterative ILP approach, with notable reductions in the average solution time.     

Ranking in integer linear fractional programming problems

An algorithm is developed which ranks the feasible solutions of an integer fractional programming problem in decreasing order of the objective function values.

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Zusammenfassung Es wird ein Algorithmus angegeben, der die zulässigen Lösungen eines ganzzahligen Quotientenprogrammes nach fallenden Zielfunktionswerten liefert.

     

The monic integer transfinite diameter

We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval . The monic integer transfinite diameter is defined as the infimum of all such supremums. We show that if has length , then .

We make three general conjectures relating to the value of for intervals of length less than . We also conjecture a value for where . We give some partial results, as well as computational evidence, to support these conjectures.

We define functions and , which measure properties of the lengths of intervals with on either side of . Upper and lower bounds are given for these functions.

We also consider the problem of determining when is a Farey interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning this value is true for an infinite family of Farey intervals.

     

An integral transformation for integer programming problems

It is shown that every integer programming problem can be transformed into an equivalent integer program with free variables in polynomial time. The transformation is advantageous because the equivalent problem it generates can be solved very easily in some restricted cases.     

Remarks on a conjecture on certain integer sequences

Summary The periodicity of sequences of integers <InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>(a_{n})_{n\in\mathbb Z}$ satisfying the inequalities <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> 0 \le a_{n-1}+\lambda a_n +a_{n+1} < 1 \ (n \in {\mathbb Z}) $$ is studied for real $ \lambda $ with $|\lambda|< 2$. Periodicity is proved in case $ \lambda $ is the golden ratio; for other values of $ \lambda $ statements on possible period lengths are given. Further interesting results on the morphology of periods are illustrated. The problem is connected to the investigation of shift radix systems and of Salem numbers.