整系数多项式有理根的一个新求法

求整系数多项式的有理根 ,现行的高等数学书中只有一个经典的方法 ,本文给出了第二个有趣的简捷方法 ,这种方法的主要过程只需进行简单的算术运算     

关于整系数多项式有理根求法的注记

现行高等代数教材给出了求整系数多项式有理根的经典方法 ,周仲旺近日撰文又给出了一个新方法 ,称其“要比经典的方法有趣简捷”,但没有给出两个方法运算量的定量分析与比较 .本文先对经典方法从数学原理和算法设计两个方面作较详细明确的描述 ;再给出经典算法与周方法运算量的定量分析 ,比较的结果是周方法运算量比经典算法运算量多得多 .     

整系数多项式有理根一个新求法的再探讨

设f (x)为整系数多项式,α为有理数,对n个不同的整数t1,…,tn,gα(tk) =f (tk)tk-α都是整数,那么α是f (x)的根的充要条件是f (t) =∑ni=1∏1≤j≤nj≠it-tjti-tjgα(ti) ( t∈Z) .     

The Monotonicity Problem in Finding Roots of Polynomials by Kuhn's Algorithm

In this paper the problem proposed by Kuhn on the presence of a monotonicity property related to the Kuhn's algorithm for finding roots of a polynomials is solved in the affirmative. Furthermore, an estimate of the threshold number D in the above-mentioned monotonicity problem expressed in terms of the complex coefficients of the polynomial is obtained.     

Generalized Trajectory Methods for Finding Multiple Extrema and Roots of Functions

Two generalized trajectory methods are combined to provide a novel and powerful numerical procedure for systematically finding multiple local extrema of a multivariable objective function. This procedure can form part of a strategy for global optimization in which the greatest local maximum and least local minimum in the interior of a specified region are compared to the largest and smallest values of the objective function on the boundary of the region. The first trajectory method, a homotopy scheme, provides a globally convergent algorithm to find a stationary point of the objective function. The second trajectory method, a relaxation scheme, starts at one stationary point and systematically connects other stationary points in the specified region by a network of trjectories. It is noted that both generalized trajectory methods actually solve the stationarity conditions, and so they can also be used to find multiple roots of a set of nonlinear equations.     

On the Roots of Domination Polynomials

The domination polynomial of a graph G of order n is the polynomial ${D(G, x) = \sum_{i=\gamma(G)}^{n} d(G, i)x^i}$ where d(G, i) is the number of dominating sets of G of size i, and γ(G) is the domination number of G. We investigate here domination roots, the roots of domination polynomials. We provide an explicit family of graphs for which the domination roots are in the right half-plane. We also determine the limiting curves for domination roots of complete bipartite graphs. Finally, we prove that the closure of the roots is the entire complex plane.     

多项式的根的一个性质的判定

文献[1]在讨论多项式型的函数迭代方程的局部解析解的存在性时涉及到了多项式的根的一个性质.本文给出了判定该性质是否成立的一个简洁的条件,证明了多项式λ_nzⁿ+…+λ₂z²+λ₁z+λ₀有一个根α满足inf{｜λ_nα^nm+…+λ₂a^2m+λ₁α^m+λ₀｜：m=2,3,…｝>0当且仅当如下两个条件之中至少有一个成立：(i)该多项式有一个根β满足｜β｜>1;(ii)该多项式有一个根β满足｜β｜<1,且λ₀≠0.     

On the Roots of Expected Independence Polynomials

The independence polynomial of a (finite) graph is the generating function for the number of independent sets of each cardinality. Assuming that each possible edge of a complete graph of order n is independently operational with probability p, we consider the expected independence polynomial. We show here that for all fixed , the expected independence polynomials of complete graphs have all real, simple roots.     

Notes on the Roots of Ehrhart Polynomials

We determine lattice polytopes of smallest volume with a given number of interior lattice points. We show that the Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n², where n is the dimension. This improves on the previously best known bound n and complements a recent result of Braun where it is shown that the norm of a root of a Ehrhart polynomial is at most of order n². For the class of 0-symmetric lattice polytopes we present a conjecture on the smallest volume for a given number of interior lattice points and prove the conjecture for crosspolytopes. We further give a characterisation of the roots of Ehrhart polyomials in the three-dimensional case and we classify for n ≤ 4 all lattice polytopes whose roots of their Ehrhart polynomials have all real part -1/2. These polytopes belong to the class of reflexive polytopes.     

Finding a Cluster of Zeros of Univariate Polynomials

A method to compute an accurate approximation for a zero cluster of a complex univariate polynomial is presented. The theoretical background on which this method is based deals with homotopy, Newton's method, and Rouché's theorem. First the homotopy method provides a point close to the zero cluster. Next the analysis of the behaviour of the Newton method in the neighbourhood of a zero cluster gives the number of zeros in this cluster. In this case, it is sufficient to know three points of the Newton sequence in order to generate an open disk susceptible to contain all the zeros of the cluster. Finally, an inclusion test based on a punctual version of the Rouché theorem validates the previous step. A specific implementation of this algorithm is given. Numerical experiments illustrate how this method works and some figures are displayed.     

Roots of Ehrhart Polynomials of Smooth Fano Polytopes

V. Golyshev conjectured that for any smooth polytope P with dim(P)≤5 the roots z∈ℂ of the Ehrhart polynomial for P have real part equal to −1/2. An elementary proof is given, and in each dimension the roots are described explicitly. We also present examples which demonstrate that this result cannot be extended to dimension six.     

Perturbations of Roots under Linear Transformations of Polynomials

Let be the complex vector space of all polynomials of degree at most n. We give several characterizations of the linear operators for which there exists a constant C > 0 such that for all nonconstant there exist a root u of f and a root v of Tf with . We prove that such perturbations leave the degree unchanged and, for a suitable pairing of the roots of f and Tf, the roots are never displaced by more than a uniform constant independent on f. We show that such "good" operators T are exactly the invertible elements of the commutative algebra generated by the differentiation operator. We provide upper bounds in terms of T for the relevant constants.     

On the Location of Roots of Independence Polynomials

The independence polynomial of a graph G is the function i(G, x) = _k0 i _k x ^k, where i _k is the number of independent sets of vertices in G of cardinality k. We prove that real roots of independence polynomials are dense in (–, 0], while complex roots are dense in , even when restricting to well covered or comparability graphs. Throughout, we exploit the fact that independence polynomials are essentially closed under graph composition.     

Roots of Independence Polynomials of Well Covered Graphs

Let G be a well covered graph, that is, all maximal independent sets of G have the same cardinality, and let i_k denote the number of independent sets of cardinality k in G. We investigate the roots of the independence polynomial i(G, x) = i_kx^k. In particular, we show that if G is a well covered graph with independence number , then all the roots of i(G, x) lie in in the disk |z| (this is far from true if the condition of being well covered is omitted). Moreover, there is a family of well covered graphs (for each ) for which the independence polynomials have a root arbitrarily close to –.     

On the Location of Roots of Graph Polynomials

Roots of graph polynomials such as the characteristic polynomial, the chromatic polynomial, the matching polynomial, and many others are widely studied. In this paper we examine to what extent the location of these roots reflects the graph theoretic properties of the underlying graph.     

Schanuel's Conjecture and Algebraic Roots of Exponential Polynomials

In this article, I will prove that assuming Schanuel's conjecture, an exponential polynomial with algebraic coefficients can have only finitely many algebraic roots. Furthermore, this proof demonstrates that there are no unexpected algebraic roots of any such exponential polynomial. This implies a special case of Shapiro's conjecture: if p(x) and q(x) are two exponential polynomials with algebraic coefficients, each involving only one iteration of the exponential map, and they have common factors only of the form exp (g) for some exponential polynomial g, then p and q have only finitely many common zeros.     

最优稳定多项式的根及误差常数

本文研究了最优稳定多项式的根的一些性质 ,并证明了误差常数总是正的 ,而且 ,对给定的阶 ,误差常数随着多项式次数的增加而减小 .     

A Comparative Study of Numerical Methods for Moving Boundary Problems

Present address: Mathematics Division, Shell Research Ltd, Thomton Research Centre, P.O. Box 1, Chester, Cheshire. This paper describes, develops and compares several viable methodsfor the numerical solution of one (space) dimensional, movingboundary (Stefan) problems. The methods are examined for computationalefficiency and accuracy, programming complexity and ease ofgeneralization to more than one (space) dimension, complicatedmoving boundary conditions, rapidly-moving boundaries and nonlineargoverning equations or fixed boundary conditions.