scholarly journals Bounding the Number of Common Zeros of Multivariate Polynomials and Their Consecutive Derivatives

2018 ◽  
Vol 28 (2) ◽  
pp. 253-279
Author(s):  
O. GEIL ◽  
U. MARTÍNEZ-PEÑAS
Keyword(s):  
Upper Bound ◽  
Existence And Uniqueness ◽  
Combinatorial Nullstellensatz ◽  
Finite Collection ◽  
Multivariate Polynomials ◽  
Grobner Basis ◽  
Evaluation Codes ◽  
Number Of Zeros ◽  
Interpolating Polynomials ◽  
Common Zeros

We upper-bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.

scholarly journals Existence and uniqueness of analytic solutions of the Shabat equation

2005 ◽  
Vol 2005 (8) ◽  
pp. 855-862 ◽  
Author(s):  
Eugenia N. Petropoulou
Keyword(s):  
Upper Bound ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Analytic Solution ◽  
Sufficient Conditions ◽  
Analytic Solutions ◽  
Initial Condition ◽  
Initial Value ◽  
First Derivative

Sufficient conditions are given so that the initial value problem for the Shabat equation has a unique analytic solution, which, together with its first derivative, converges absolutely forz∈ℂ:|z|<T,T>0. Moreover, a bound of this solution is given. The sufficient conditions involve only the initial condition, the parameters of the equation, andT. Furthermore, from these conditions, one can obtain an upper bound forT. Our results are in consistence with some recently found results.

scholarly journals Sharp Bound of the Number of Zeros for a Liénard System with a Heteroclinic Loop

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Junning Cai ◽  
Minzhi Wei ◽  
Guoping Pang
Keyword(s):  
Limit Cycles ◽  
Upper Bound ◽  
Sharp Bound ◽  
Abelian Integral ◽  
Heteroclinic Loop ◽  
Liénard System ◽  
Algebraic Criterion ◽  
Number Of Zeros ◽  
Lienard System ◽  
Nilpotent Saddle

In the presented paper, the Abelian integral I h of a Liénard system is investigated, with a heteroclinic loop passing through a nilpotent saddle. By using a new algebraic criterion, we try to find the least upper bound of the number of limit cycles bifurcating from periodic annulus.

scholarly journals Generalized Iterated Function Systems Containing Functions of Integral Type

2018 ◽  
Vol 7 (3.31) ◽  
pp. 126
Author(s):  
Minirani S ◽  
. .
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Product Space ◽  
Iterated Function System ◽  
Complete Metric Space ◽  
Iterated Function Systems ◽  
Function System ◽  
Finite Collection ◽  
Integral Type ◽  
Iterated Function

A finite collection of mappings which are contractions on a complete metric space constitutes an iterated function system. In this paper we study the generalized iterated function system which contain generalized contractions of integral type from the product space . We prove the existence and uniqueness of the fixed point of such an iterated function system which is also known as its attractor. 

scholarly journals Parameterized Codes over Cycles

2013 ◽  
Vol 21 (3) ◽  
pp. 241-256 ◽  
Author(s):  
Manuel González Sarabia ◽  
Joel Nava Lara ◽  
Carlos Rentería Marquez ◽  
Eliseo Sarmíento Rosales
Keyword(s):  
Upper Bound ◽  
Minimum Distance ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Evaluation Codes ◽  
Singleton Bound ◽  
Odd Cycles

AbstractIn this paper we will compute the main parameters of the parameterized codes arising from cycles. In the case of odd cycles the corresponding codes are the evaluation codes associated to the projective torus and the results are well known. In the case of even cycles we will compute the length and the dimension of the corresponding codes and also we will find lower and upper bounds for the minimum distance of this kind of codes. In many cases our upper bound is sharper than the Singleton bound.

scholarly journals Preventing extinction in Rastrelliger brachysoma using an impulsive mathematical model

2021 ◽  
Vol 7 (1) ◽  
pp. 1-24
Author(s):  
Din Prathumwan ◽  
◽  
Kamonchat Trachoo ◽  
Wasan Maiaugree ◽  
Inthira Chaiya ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Periodic Solution ◽  
Population Density ◽  
Upper Bound ◽  
Existence And Uniqueness ◽  
Periodic Behavior ◽  
Toxic Environment ◽  
Pacific Mackerel ◽  
The Stability ◽  
Theoretical Results

<abstract><p>In this paper, we proposed a mathematical model of the population density of Indo-Pacific mackerel (<italic>Rastrelliger brachysoma</italic>) and the population density of small fishes based on the impulsive fishery. The model also considers the effects of the toxic environment that is the major problem in the water. The developed impulsive mathematical model was analyzed theoretically in terms of existence and uniqueness, positivity, and upper bound of the solution. The obtained solution has a periodic behavior that is suitable for the fishery. Moreover, the stability, permanence, and positive of the periodic solution are investigated. Then, we obtain the parameter conditions of the model for which Indo-Pacific mackerel conservation might be expected. Numerical results were also investigated to confirm our theoretical results. The results represent the periodic behavior of the population density of the Indo-Pacific mackerel and small fishes. The outcomes showed that the duration and quantity of fisheries were the keys to prevent the extinction of Indo-Pacific mackerel.</p></abstract>

scholarly journals ON THE NUMBER OF ZEROS OF THE ABELIAN INTEGRALS FOR A CLASS OF PERTURBED LIÉNARD SYSTEMS

2007 ◽  
Vol 17 (09) ◽  
pp. 3281-3287
Author(s):  
TONGHUA ZHANG ◽  
YU-CHU TIAN ◽  
MOSES O. TADÉ
Keyword(s):  
Algebraic Structure ◽  
Upper Bound ◽  
Abelian Integrals ◽  
Hilbert’S 16Th Problem ◽  
Number Of Zeros ◽  
Liénard Systems ◽  
Hilbert's 16Th Problem

Addressing the weakened Hilbert's 16th problem or the Hilbert–Arnold problem, this paper gives an upper bound B(n) ≤ 7n + 5 for the number of zeros of the Abelian integrals for a class of Liénard systems. We proved the main result using the Picard–Fuchs equations and the algebraic structure of the integrals.

EXPLICIT ZERO-COUNTING THEOREM FOR HECKE–LANDAU ZETA-FUNCTIONS

2017 ◽  
Vol 95 (3) ◽  
pp. 400-411 ◽  
Author(s):  
MACIEJ GRZEŚKOWIAK
Keyword(s):  
Upper Bound ◽  
Zeta Functions ◽  
Number Of Zeros

We give an explicit upper bound for the number of zeros of Hecke–Landau zeta-functions in a rectangle.

scholarly journals On approximation by trigonometric Lagrange interpolating polynomials

1989 ◽  
Vol 40 (3) ◽  
pp. 425-428 ◽  
Author(s):  
T.F. Xie ◽  
S.P. Zhou
Keyword(s):  
Upper Bound ◽  
Best Approximation ◽  
Arbitrary Degree ◽  
Interpolating Polynomials ◽  
Lagrange Interpolating Polynomials

It is well-known that the approximation to f(x) ∈ C2π, by nth trigonometric Lagrange interpolating polynomials with equally spaced nodes in C2π, has an upper bound In(n)En(f), where En(f) is the nth best approximation of f(x). For various natural reasons, one can ask what might happen in Lp space? The present paper indicates that the result about the trigonometric Lagrange interoplating approximation in Lp space for 1 < p < ∞ may be “bad” to an arbitrary degree.

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