We introduce the notions of alternating roots of polynomials and alternating
polynomials over a Cayley-Dickson algebra, and prove a connection between the
alternating roots of a given polynomial and ...the roots of the corresponding
alternating polynomial over the Cayley-Dickson doubling of the algebra. We also
include a detailed Octave code for the computation of alternating roots over
Hamilton's quaternions.
We apply a coctel of elementary methods to the problem of finding the roots of an arbitrary polynomial. Specifically, we combine properties of the iteration z → z2 + c with rudimentary Galois theory ...in order to justify an algorithm to find the roots of a complex polynomial.
Let μ be a compactly supported probability measure on the real line. Bercovici-Voiculescu and Nica-Speicher proved the existence of a free convolution power
for any real
. The purpose of this short ...note is to give a formal proof of an elementary description of
in terms of polynomials and roots of their derivatives. This bridge allows us to switch back and forth between free probability and the asymptotic behavior of polynomials.
The topology of Arnold–Liouville level sets of the real forms of the complex generic Neumann system depends indirectly on the positions of the roots of the special polynomial US(λ). For certain ...polynomials, the existence and positions of the real roots, according to the suitable parameters of the system, is not obvious.
In the paper, a method for checking the existence and positions of the real roots of the polynomials US(λ) is given. The method and algorithm are based on searching of a positive solution of a system of linear equations. In case n=2, we provide a complete solution to the problem of existence of real roots for all special polynomials and determine the topology of the Arnold–Liouville level sets.
In 1, Brändén, Krasikov, and Shapiro study root location preservation properties of finite difference operators. They then conjecture a characterization of linear finite difference operators on ...polynomials of fixed degree which preserve root mesh properties, via a natural polynomial convolution. We prove this conjecture using two methods. The first develops a novel connection between the additive (Walsh) and multiplicative (Grace-Szegö) convolutions, which can be generically used to transfer results from multiplicative to additive. We then use this to transfer an analogous result, due to Lamprecht 4, which demonstrates logarithmic root mesh preservation properties of a certain q-multiplicative convolution operator. The second method proves the result directly using a modification of Lamprecht's proof of the logarithmic root mesh result. We present his original argument in a streamlined fashion and then make the appropriate alterations to apply it to the additive case.
Let K be an algebraically closed field with an absolute value. This note gives a simple proof of the classical result that the roots of a polynomial with coefficients in K are continuous functions of ...the coefficients of the polynomial.
We consider paths of steepest descent, in the complex plane, for the norm of a non-constant one variable polynomial f. We show that such paths, starting from a zero of the logarithmic derivative of f ...and ending in a root of f, draw a tree in the complex plane, and we give an upper bound estimate on their lengths. In some cases, we obtain a finer estimate that depends only on the set of roots of f, not on their multiplicity, and we wonder if this can be done in general. We also extend this question to finite Blaschke products for the unit disk.
Coulson integral formula was initiated in the research of energy of graphs in 1940 by Coulson, which provides an efficient way to calculate the energy of graphs without knowing any eigenvalues of ...graphs. Recently, Qiao et al. generalized the concept of energy from graphs to polynomials, which is referred to as the general energy of polynomials. For a polynomial ϕ(z) and a nonzero real number α, the general energy of polynomial ϕ(z), denoted by Eα(ϕ), is defined as the sum of αth powers of absolute values of all (nonzero) roots of ϕ(z), i.e., Eα(ϕ)=∑zk≠0|zk|α, where zk represents the nonzero roots of ϕ(z). In a series of three publications by Qiao et al., they gave some beautiful Coulson-type integral formulas for Eα(ϕ) when the roots of ϕ(z) are all nonnegative numbers for any rational number α and any irrational number α with 0<|α|<1, and when the roots of ϕ(z) are all real numbers for any rational number α and any irrational number α with 0<|α|<2. For the remaining α, their integral formulas are invalid. In this paper, we set up a research procedure covering all the nonzero real numbers α. That is to say, it not only combines all the Coulson-type integral formulas established by Qiao et al., but also involves in all the remaining unsolved α. As an extension and generalization of known integral formulas, the Coulson-type integral formulas for Eα(ϕ), when the roots of ϕ(z) are all pure imaginary numbers, are also presented, for any nonzero real number α. Moreover, we also give an alternative form of integral formulas expressed by the coefficients of polynomials.
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