Analysis of the distribution of roots of a polynomial using a generalized Routh scheme

1996 ◽  
Vol 82 (3) ◽  
pp. 3479-3487
Author(s):  
A. T. Barabanov
Keyword(s):  
Distribution Of Roots ◽  
Roots Of A Polynomial

scholarly journals Statistical Distribution of Roots of a Polynomial Modulo Primes III

2017 ◽  
Vol 7 (1) ◽  
pp. 115
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Prime Number ◽  
Linear Relation ◽  
Statistical Distribution ◽  
Ordinary Sense ◽  
Complex Roots ◽  
Distinct Integer ◽  
Distribution Of Roots ◽  
Roots Of A Polynomial ◽  
Expected Density

Let $f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ $(a_{n-1},\dots,a_0\in\mathbb Z)$ be a polynomial with complex roots $\alpha_1,\dots,\alpha_n$ and suppose that a linear relation over $\mathbb Q$ among $1,\alpha_1,\dots,\alpha_n$ is a  multiple of $\sum_i\alpha_i+a_{n-1}=0$ only. For a prime number $p$ such that $f(x)\bmod p$ has $n$ distinct integer  roots $0<r_1<\dots<r_n<p$, we proposed in a previous paper a conjecture that the sequence of points $(r_1/p,\dots,r_n/p)$ is equi-distributed in some sense. In this paper, we show that it implies the equi-distribution of the sequence of $r_1/p,\dots,r_n/p$ in the ordinary sense and give the expected density of primes satisfying $r_i/p<a$ for a fixed suffix $i$ and $0<a<1$.

scholarly journals Statistical Distribution of Roots of a Polynomial Modulo Primes II

2017 ◽  
Vol 12 (1) ◽  
pp. 109-122
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Irreducible Polynomial ◽  
Statistical Distribution ◽  
Distribution Of Roots ◽  
Roots Of A Polynomial

Abstract Continuing the previous paper, we give several data on the distribution of roots modulo primes of an irreducible polynomial, and based on them, we propose problems on the distribution.

Spatial Distribution of Roots across Three Dryland Ecosystems and Plant Functional Types

2019 ◽  
Vol 79 (2) ◽  
pp. 159 ◽  
Author(s):  
Jessica G. Swindon ◽  
William K. Lauenroth ◽  
Daniel R. Schlaepfer ◽  
Ingrid C. Burke
Keyword(s):  
Spatial Distribution ◽  
Plant Functional Types ◽  
Functional Types ◽  
Dryland Ecosystems ◽  
Distribution Of Roots

scholarly journals Hopf bifurcation of a delayed worm model with two latent periods

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Juan Liu ◽  
Zizhen Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Sensor Network ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Worm Model ◽  
Distribution Of Roots ◽  
Influential Parameters

Abstract We investigate a delayed epidemic model for the propagation of worm in wireless sensor network with two latent periods. We derive sufficient conditions for local stability of the worm-induced equilibrium of the system and the existence of Hopf bifurcation by regarding different combination of two latent time delays as the bifurcation parameter and analyzing the distribution of roots of the associated characteristic equation. In particular, we investigate the direction and stability of the Hopf bifurcation by means of the normal form theory and center manifold theorem. To verify analytical results, we present numerical simulations. Also, the effect of some influential parameters of sensor network is properly executed so that the oscillations can be reduced and removed from the network.

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.

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