Root neighborhoods of a polynomial

CHAPTER ELEVEN - ROOTS OF POLYNOMIALS

CHAPTER TEN - ROOTS OF POLYNOMIALS

Let Σ denote the set of all seriesof complex numbers. By a ‘summability method’, say A, we mean a function from some subset (the set of ‘A -summable series’) of Σ into the set of complex numbers. We will use the language generally associated with this definition, and will take for granted the case in which A is (C, 1). A summability method A will be called linear if, whenever a, b are A -summable, then so is λa + μb (where λ, μ are any complex constants) and if the. A -sums of a, b, λa + μb are then related in the natural way. We call A regular if, whenever a converges to σ, it is A -summable to σ. If A is a regular summability method, then any condition P on the series (1) will be called a Tauberian condition for A if any A -summable series which satisfies P is convergent.

The rapid development of mathematics is largely due to the creation of complex numbers. Although complex numbers are just strange and seemingly meaningless notations for most people, complex numbers play important roles in engineering fields. A better understanding of complex numbers' geometric and algebraic structure enables us to study complex analysis better. Complex numbers convince scientists that our world is magical, full of wonderful insights, and even miraculous. In this paper, I first review several basic properties of complex numbers. The set of complex numbers is a group under the operation of the multiplication and under the operation of addition. Then I visualize complex numbers by building a bijective connection between complex numbers and points on the complex plane. I also give several alternative forms of complex numbers, namely the trigonometric and general forms. By invoking the arithmetic properties of complex numbers, I prove two trigonometric identities.

Finding all n roots of the general polynomial over the complex field can be difficult theoretically. Therefore, presented here is an approach that attempts to combine algebraic theory (factor theorem) with multi‐stage Monte Carlo optimization to find the n roots of the general polynomial. An example of degree twenty‐five is presented. Rounding error problems for polynomials of degree fifty or over are discussed also.

On the Expected Number of Real Zeros of Random Polynomials. II. Coefficients With Non-Zero Means

Given a complex number λ of modulus 1, we show that the bifurcation locus of the one parameter family {f b (z)=λz+b z 2+z 3} b ∈ ℂ contains quasi-conformal copies of the quadratic Julia set J(λz+z 2). As a corollary, we show that when the Julia set J(λz+z 2) is not locally connected (for example when z↦λz+z 2 has a Cremer point at 0), the bifurcation locus is not locally connected. To our knowledge, this is the first example of complex analytic parameter space of dimension 1, with connected but non-locally connected bifurcation locus. We also show that the set of complex numbers λ of modulus 1, for which at least one of the parameter rays has a non-trivial accumulation set, contains a dense G δ subset of S 1.

In 1985, Barnsley and Harrington defined a ‘Mandelbrot Set’${\mathcal{M}}$for pairs of similarities: this is the set of complex numbers$z$with$0<|z|<1$for which the limit set of the semigroup generated by the similarities$$\begin{eqnarray}x\mapsto zx\quad \text{and}\quad x\mapsto z(x-1)+1\end{eqnarray}$$is connected. Equivalently,${\mathcal{M}}$is the closure of the set of roots of polynomials with coefficients in$\{-1,0,1\}$. Barnsley and Harrington already noted the (numerically apparent) existence of infinitely many small ‘holes’ in${\mathcal{M}}$, and conjectured that these holes were genuine. These holes are very interesting, since they are ‘exotic’ components of the space of (2-generator) Schottky semigroups. The existence of at least one hole was rigorously confirmed by Bandt in 2002, and he conjectured that the interior points are dense away from the real axis. We introduce the technique oftrapsto construct and certify interior points of${\mathcal{M}}$, and use them to prove Bandt’s conjecture. Furthermore, our techniques let us certify the existence of infinitely many holes in${\mathcal{M}}$.

This paper concerns the location of roots of polynomials with respect to the principal coordinate axes (referred to generally as the Routh–Hurwitz criteria and the Sturm criteria). It provides a new proof of the results established by Wall and Frank which relate the number of positive and negative coefficients in a continued fraction to the number of roots of a polynomial in the left half plane and right half plane. It extends these results to locate the number of real roots of real polynomials on the positive or negative real axis. The paper also provides some miscellaneous criteria so that selected parts of a polynomial may be analyzed in order to determine the number of roots in a half plane.

High dimensional array data, tensor data, is becoming important in recent days. Then maximal rank of tensors is important in theory and applications. In this paper we consider the maximal rank of 3 tensors. It can be attacked from various viewpoints, however, we trace the method of Atkinson-Stephens(1979) and Atkinson-Lloyd(1980). They treated the problem in the complex field, and we will present various bounds over the real field by proving several lemmas and propositions, which is real counterparts of their results.

A chromatic root is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008. The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

Recent Advances on Determining the Number of Real Roots of Parametric Polynomials

Tricomplex numbers are a generalization of bicomplex numbers. In this paper, we detail a technique for finding the roots of tricomplex polynomials. We generalized then the process to multicomplex polynomials. We first give an idempotent-representation of tricomplex numbers and reduce the working method to complex polynomials. We give examples to illustrate the different situations. Finally, for a multicomplex polynomial, we explain a reduction process ending to search roots in the complex field. Combining these give the roots for multicomplex polynomials.

A refined realization theorem in the context of the Schur–Szegő composition

With changing strong nonlinear Duffing equation subject to harmonic excitation with damping in complex number field as an object, the amplitude frequency response equation of primary resonance of the system is obtained through parametric transformation with application of MLP method and modified method of multiple scales. In different approximation solution forms and different time scales, the two methods lead to the same amplitude frequency response equation. Thus, the two methods are mutually verifiable. Numerical analysis shows that for the strong nonlinear Duffing equation with damping in complex number field, the nonlinear stiffness coefficient is more than zero and the amplitude frequency response curve of primary resonance leans to the left, which is different from the weak nonlinear Duffing equation.Chinese books catalog: O321