不可约多项式的几个充分条件

多项式的因式分解对解多项式方程起着重要作用,因而判定一多项式是否可约是一个重要问题,本文将给出一些多项式不可约的充分条件。 定义 若f(x)∈Z[x]。(Z表所有整数构成的集合),f(x)的次数大于零,f(x)不能表示成两个次数大于零的整系数多项式的乘积,则称f(x)在Z上不可约,否则称f(x)在Z上可约。

关于Schur猜想的研究

设ｍ，ｎ是正整数，ａ１，ａ２，…，ａｎ是不同的奇数。证明了当ｍ＞１，且ｎ＝２ｉｎ１（ｎ１为奇数）时，Ｓｃｈｕｒ多项式ｆ（ｘ）＝（ｘ－ａ１）２ｍ（ｘ－ａ２）２ｍ…（ｘ－ａｎ）２ｍ＋１必有一次数不小于２ｍ＋ｉ的不可约因式。该结果部分地证实了Ｓｃｈｕｒ猜想，并使Ｓｃｈｕｒ猜想以前的结果得到了发展。

Low latency systolic multipliers for finite field GF（2m） based on irreducible polynomials

Systolic implementation of multiplication over GF(2m) is usually very efficient in area-time complexity,but its latency is usually very large.Thus,two low latency systolic multipliers over GF(2m) based on general irreducible polynomials and irreducible pentanomials are presented.First,a signal flow graph(SFG) is used to represent the algorithm for multiplication over GF(2m).Then,the two low latency systolic structures for multiplications over GF(2m) based on general irreducible polynomials and pentanomials are presented from the SFG by suitable cut-set retiming,respectively.Analysis indicates that the proposed two low latency designs involve at least one-third less area-delay product when compared with the existing designs,To the authors' knowledge,the time-complexity of the structures is the lowest found in literature for systolic GF(2m) multipliers based on general irreducible polynomials and pentanomials.The proposed low latency designs are regular and modular,and therefore they are suitable for many time critical applications.

New Pseudorandom Number Generator Artin-Sc hreier Tower for p = 5

The standard method to construct a finite field requires a primitive irreducible polynomial of a given degree. Therefore, it is difficult to apply for the construction of huge finite fields. To avoid this problem, we propose a new method to construct huge finite fields with the characteristic p = 5 by using an Artin-Schreier tower. Utilizing the recursive basis of the Artin-Schreier tower, we define a nmltiplication algorithm The algorithm can explicitly calculate the multiplication of two elements on the top finite field of this tower, without any primitive element. We also define a linear recurrence equation as an application, which produces a sequence of numbers, and call the new pseudorandom number generator Abstract Syntax Tree （AST） for p = 5. The experircental results show that our new pseudorandom number generator can produce a sequence of numbers with a long period.

Some Structures of Irreducible Polynomials over a Unique Factorization Domain R

In this paper, we give the conception of implicit congruence and nonimplicit congruence in a unique factorization domain R and establish some structures of irreducible polynomials over R . A classical result, Eisenstein′s criterion, is generalized.

A CRITERION FOR TESTING WHETHER A DIFFERENCE IDEAL IS PRIME

This paper presents a criterion for testing the irreducibility of a polynomial over an algebraicextension field.Using this criterion and the characteristic set method,the authors give a criterion fortesting whether certain difference ascending chains are strong irreducible,and as a consequence,whetherthe saturation ideals of these ascending chains are prime ideals.

A C~*-ALGEBRA APPROACHTO THE IRREDUCIBILITY OFCOWEN-DOUGLAS OPERATORS

The authors consider the irreducibility of the Cowen-Douglas operator T. It is proved that T is irreducible iff the unital C*-algebra generated by some non-zero blocks in the decomposition of T with respect to (Ker Tn+1 Ker Tn) is M.(C).

Periods of polynomials over a Galois ring

The period of a monic polynomial over an arbitrary Galois ring GR(pe,d) is theoretically determined by using its classical factorization and Galois extensions of rings. For a polynomial f(x) the modulo p remainder of which is a power of an irreducible polynomial over the residue field of the Galois ring, the period of f(x) is characterized by the periods of the irreducible polynomial and an associated polynomial of the form (x-1)m+pg(x). Further results on the periods of such associated polynomials are obtained, in particular, their periods are proved to achieve an upper bound value in most cases. As a consequence, the period of a monic polynomial over GR(pe,d) is equal to pe-1 times the period of its modulo p remainder polynomial with a probability close to 1, and an expression of this probability is given.

L-Factor of Irreducible χ_1×χ_2■σ

The L-factor of irreducible x1 × x2 ×｜ σdefined by Piatetski-Shapiro is computed by using non-split Bessel functional.

Optimality conditions for sparse nonlinear programming

The sparse nonlinear programming （SNP） is to minimize a general continuously differentiable func- tion subject to sparsity, nonlinear equality and inequality constraints. We first define two restricted constraint qualifications and show how these constraint qualifications can be applied to obtain the decomposition properties of the Frechet, Mordukhovich and Clarke normal cones to the sparsity constrained feasible set. Based on the decomposition properties of the normal cones, we then present and analyze three classes of Karush-Kuhn- Tucker （KKT） conditions for the SNP. At last, we establish the second-order necessary optimality condition and sufficient optimality condition for the SNP.

Quantification of Symmetry

Symmetry is conventionally described in a polarized manner that the system is either completely symmetric or completely asymmetric.Using group theoretical approach to overcome this dichotomous problem,we introduce the degree of symmetry(DoS) as a non-negative continuous number ranging from zero to unity.Do S is defined through an average of the fidelity deviations of Hamiltonian or quantum state over its transformation group G,and thus is computable by making use of the completeness relations of the irreducible representations of G.The monotonicity of Do S can effectively probe the extended group for accidental degeneracy while its multi-valued natures characterize some(spontaneous) symmetry breaking.