On the Coefficients and Zeros of a Polynomial

Letp(z)=1+∑ⁿ_j=1 b_jz^jbe a complex polynomial. Two theorems on the coefficients and zeros ofp(z) are proved in this paper.     

Graeffe's, Chebyshev-like, and Cardinal's Processes for Splitting a Polynomial into Factors

Numerical splitting of a real or complex univariate polynomial into factors is the basic step of the divide-and-conquer algorithms for approximating complex polynomial zeros. Such algorithms are optimal (up to polylogarithmic factors) and are quite promising for practical computations. In this paper, we develop some new techniques, which enable us to improve numerical analysis, performance, and computational cost bounds of the known splitting algorithms. In particular, we study a Chebyshev-like modification of Graeffe's lifting iteration (which is a basic block of the splitting algorithms, as well as of several other known algorithms for approximating polynomial zeros), analyze its numerical performance, compare it with Graeffe's, prove some results on numerical stability of both lifting processes (that is, Graeffe's and Chebyshev-like), study their incorporation into polynomial root-finding algorithms, and propose some improvements of Cardinal's recent effective technique for numerical splitting of a polynomial into factors. Our improvement relies, in particular, on a modification of the matrix sign iteration, based on the analysis of some conformal mappings of the complex plane and of techniques of recursive lifting/recursive descending. The latter analysis reveals some otherwise hidden correlations among Graeffe's, Chebyshev-like, and Cardinal's iterative processes, and we exploit these correlations in order to arrive at our improvement of Cardinal's algorithm. Our work may also be of some independent interest for the study of applications of conformal maps of the complex plane to polynomial root-finding and of numerical properties of the fundamental techniques for polynomial root-finding such as Graeffe's and Chebyshev-like iterations.     

On Computing Zeros of a Bivariate Bernstein Polynomial

1.IntroductionIn[6]and[4],theproblemoffindingtheintersectionofacubicB6zierpatchandaplanewasconsidered.[6]consideredarectangular,and[41atriangularpatch.SincetheBernsteinoperatorB.:f-Bn(f)preserveslinearfunctions,theproblemwassimplifiedtothecomputationofzerosofabivariateBernsteinpolynomialB.(f).BothpaPersproducedsimpleandefficientcomputationalalgorithms.Itisbaseduponthefollowingidea:determinethepointswhereinsidethesupportthetopologyofzerosofB.(f)changes.Thiswasdonebyrestrictingthebivariatepo…     

Jones多项式的零点

利用数论理论证明了纽结的Jones多项式仅有可能的有理根是O,而链环的Jones多项式仅有可能的有理根是0和-1.给出了作为Jones多项式根的所有可能单位根,以及所有可能的具有平凡Mahler测度的Jones多项式.最后指出了交叉数不超过11的纽结中,只有4_1,8_9,9_(42),K11n19的Jones多项式具有平凡的Mahler测度,从而回答了林晓松提出的关于Mahler测度的一个问题.     

Jones多项式的零点

利用数论理论证明了纽结的Jones多项式仅有可能的有理根是0,而链环的Jones多项式仅有可能的有理根是0和-1.给出厂作为Jones多项式根的所有可能单位根,以及所有可能的具有平凡Mahler测度的Jones多项式.最后指出了交叉数不超过11的纽结中,只有41,89,942,K11n19的Jones多项式具有平凡的Mahler测度,从而回答了林晓松提出的关于Mahler测度的一个问题.     

On the Splitting Ring of a Polynomial

Let f(Z) = Zⁿ ? a₁Z^n?1 + … + (?1)^n?1a_n?1Z + (?1)ⁿa_n be a monic polynomial with coefficients in a ring R with identity, not necessarily commutative. We study the ideal I_f of R[X₁,…, X_n] generated by σ_i(X₁,…, X_n) ? a_i, where σ₁,…, σ_n are the elementary symmetric polynomials, as well as the quotient ring R[X₁,…, X_n]/I_f.     

Preserving Zeros of a Polynomial

We study nonlinear surjective mappings on ? _n () and its subsets, which preserve the zeros of some fixed polynomials in noncommuting variables.     

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.     

On Zeros of Real Trigonometric Sums

The problem of calculating the number of zeros of a real trigonometric sum of an arbitrary form on a given interval is considered. Upper and lower bounds for this number are obtained by using the argument principle and are illustrated by examples.     

On the Distribution of the Derivative Zeros of a Complex Polynomial

We consider the class S(n) of all complex polynomials of degree n > 1 having all their zeros in the closed unit disk ē. By S(n,β) we denote the subclass of p ? S(n) vanishing in the prescribed point β ? ē. For an arbitrary point α ? C and p ? S(n,β) let |p| _α be the distance of α and the set of zeros of P'. Then there exists some P ? S(n,β) with maximal |P|α. We give an estimation for the number of zeros of P on |z| = 1$ resp. P' on $ |z-α| = |P| _α .     

A New Method for Real Root Isolation of Univariate Polynomials

A new algorithm for real root isolation of univariate polynomials is proposed, which is mainly based on exact interval arithmetic and bisection method. Although exact interval arithmetic is usually supposed to be inefficient, our algorithm is surprisingly fast because the termination condition of our algorithm is different from those of existing algorithms which are mostly based on Descartes’ rule of signs or Vincent’s theorem and we decrease the times of Taylor shifts in some cases. We test our algorithm on a large number of examples from the literature and report the performance.      

多项式零点保持线性映射

设H是维数大于2的复Hilbert空间,β(H)代表H上所有有界线性算子全体．假定Φ是从β(H)到其自身的弱连续线性双射．我们证明了映射Φ满足对所有的A,B∈β(H),AB=BA~*蕴涵Φ(A)Φ(B)=Φ(B)Φ(A)~*当且仅当存在非零实数c和酉算子U∈(?)(H),使得Φ(A)=cUAU~*对所有的A∈β(H)成立．     

The Zeros and the Final Value of a Polynomial Form

ABSTRACT

A polynomial form f, is a not necessarily linear map, from an infinite module over a ring 𝔷 to a finite abelian group of exponent n satisfying some additional conditions. Denote the zeros of f by Ω_f. We show it satisfies a weak closure condition. Among all 𝔷-submodules of finite index, there is a submodule B such that |f (B)| (the order of the subset f (B)) is as small as possible. f (B) is called the final value of f and D. S. Passman asks if f (B) is necessarily a subgroup of S. This paper shows that if the degree of f ≤ 2 then the final value is a subgroup and if the form f has arbitrary degree from an finitely generated infinite abelian group, then the final value is 0.     

On the Real Zeros of Positive Semidefinite Biquadratic Forms

For a positive semidefinite biquadratic forms F in (3, 3) variables, we prove that, if F has a finite number but at least 7 real zeros 𝒵(F), then it is not a sum of squares. We show also that if F has at least 11 zeros, then it has infinitely many real zeros and it is a sum of squares. It can be seen as the counterpart for biquadratic forms as the results of Choi, Lam, and Reznick for positive semidefinite ternary sextics.

We introduce and compute some of the numbers BB _{n, m} which are set to be equal to sup |𝒵(F)| where F ranges over all the positive semidefinite biquadratic forms F in (n, m) variables such that |𝒵(F)| < ∞.

We also recall some old constructions of positive semidefinite biquadratic forms which are not sums of squares and we give some new families of examples.     

Extrema of a Real Polynomial

In this paper, we investigate critical point and extrema structure of a multivariate real polynomial. We classify critical surfaces of a real polynomial f into three classes: repeated, intersected and primal critical surfaces. These different critical surfaces are defined by some essential factors of f, where an essential factor of f means a polynomial factor of f–c ₀, for some constant c ₀. We show that the degree sum of repeated critical surfaces is at most d–1, where d is the degree of f. When a real polynomial f has only two variables, we give the minimum upper bound for the number of other isolated critical points even when there are nondegenerate critical curves, and the minimum upper bound of isolated local extrema even when there are saddle curves. We show that a normal polynomial has no odd degree essential factors, and all of its even degree essential factors are normal polynomials, up to a sign change. We show that if a normal quartic polynomial f has a normal quadratic essential factor, a global minimum of f can be either easily found, or located within the interior(s) of one or two ellipsoids. We also show that a normal quartic polynomial can have at most one local maximum.     

Polynomials with All Zeros Real and in a Prescribed Interval

We provide a characterization of the real-valued univariate polynomials that have only real zeros, all in a prescribed interval [a,b]. The conditions are stated in terms of positive semidefiniteness of related Hankel matrices.     

Zeros of Differential Polynomial f（z）f（k）（z） -- a and Its Normality

Suppose that function f(z) is transcendental and meromorphic in the plane. The aim of this work is to investigate the conditions in which differential monomials f(z)f(k)(z) takes any non-zero finite complex number infinitely times and to consider the normality relation to differential monomials f(z)f(k)(z).