Continuity of the Roots of a Polynomial

2020 ◽  
Vol 127 (4) ◽  
pp. 359-363
Author(s):  
Kenichi Hirose
Keyword(s):  
Roots Of A Polynomial

THE NUMBER OF ROOTS OF A POLYNOMIAL SYSTEM

2020 ◽  
pp. 1-10
Author(s):  
NGUYEN CONG MINH ◽  
LUU BA THANG ◽  
TRAN NAM TRUNG
Keyword(s):  
Polynomial Ring ◽  
Polynomial System ◽  
Combinatorial Nullstellensatz ◽  
Roots Of A Polynomial

Abstract Let I be a zero-dimensional ideal in the polynomial ring $K[x_1,\ldots ,x_n]$ over a field K. We give a bound for the number of roots of I in $K^n$ counted with combinatorial multiplicity. As a consequence, we give a proof of Alon’s combinatorial Nullstellensatz.

scholarly journals An Efficient Method to Find Solutions for Transcendental Equations with Several Roots

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
Rogelio Luck ◽  
Gregory J. Zdaniuk ◽  
Heejin Cho
Keyword(s):  
Null Space ◽  
Heat Conduction Problem ◽  
Hankel Matrix ◽  
Polynomial Equation ◽  
Transient Heat Conduction ◽  
Transcendental Equation ◽  
Integral Theorem ◽  
Transient Heat Conduction Problem ◽  
Cauchy’S Integral Theorem ◽  
Roots Of A Polynomial

This paper presents a method for obtaining a solution for all the roots of a transcendental equation within a bounded region by finding a polynomial equation with the same roots as the transcendental equation. The proposed method is developed using Cauchy’s integral theorem for complex variables and transforms the problem of finding the roots of a transcendental equation into an equivalent problem of finding roots of a polynomial equation with exactly the same roots. The interesting result is that the coefficients of the polynomial form a vector which lies in the null space of a Hankel matrix made up of the Fourier series coefficients of the inverse of the original transcendental equation. Then the explicit solution can be readily obtained using the complex fast Fourier transform. To conclude, the authors present an example by solving for the first three eigenvalues of the 1D transient heat conduction problem.

scholarly journals Fractional Newton-Raphson Method

2021 ◽  
Vol 8 (1) ◽  
pp. 1-13
Author(s):  
A. Torres-Hernandez ◽  
F. Brambila-Paz
Keyword(s):  
Fractional Calculus ◽  
Iterative Method ◽  
Complex Numbers ◽  
Initial Condition ◽  
The Real ◽  
Complex Roots ◽  
Newton Raphson ◽  
Roots Of A Polynomial ◽  
Raphson Method

The Newton-Raphson (N-R) method is useful to find the roots of a polynomial of degree n, with n ∈ N. However, this method is limited since it diverges for the case in which polynomials only have complex roots if a real initial condition is taken. In the present work, we explain an iterative method that is created using the fractional calculus, which we will call the Fractional Newton-Raphson (F N-R) Method, which has the ability to enter the space of complex numbers given a real initial condition, which allows us to find both the real and complex roots of a polynomial unlike the classical Newton-Raphson method.

Complex Roots of a Polynomial: 10688

2000 ◽  
Vol 107 (2) ◽  
pp. 181
Author(s):  
Ioan Tomescu ◽  
Kee-Wai Lau ◽  
O. P. Lossers ◽  
K. F. Andersen ◽  
R. J. Chapman ◽  
...  
Keyword(s):  
Complex Roots ◽  
Roots Of A Polynomial

A polynomial that is a statistical prism

2003 ◽  
Vol 87 (508) ◽  
pp. 76-85
Author(s):  
Mike Osborne ◽  
Mark Osborne
Keyword(s):  
Complex Plane ◽  
Polynomial Equation ◽  
Short Description ◽  
Financial Mathematics ◽  
Time Value Of Money ◽  
Time Value ◽  
Salient Points ◽  
The Mean ◽  
Value Of Money ◽  
Roots Of A Polynomial

The roots of a polynomial can be represented as points in the complex plane. The time value of money (TVM) equation that is commonly used in finance is a polynomial equation. (See the appendix for a short description of the TVM equation and an example of its use in finance.) In [1] and [2] it is shown that concepts from financial mathematics can be obtained from the pattern of the roots of the TVM equation. The concepts are given in terms of distances between the roots and other salient points in the plane. This note shows that this particular polynomial, and the technique, can be applied more generally. When a series of data is fed into the coefficients of the polynomial, the mean and standard deviation of the data are seen in the complex plane as combinations of distances between the roots and other salient points. The results are aesthetically pleasing as well as mathematically interesting.

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