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 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 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 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.

scholarly journals On Zeros of a Polynomial in a Finite Grid

2018 ◽  
Vol 27 (3) ◽  
pp. 310-333 ◽  
Author(s):  
ANURAG BISHNOI ◽  
PETE L. CLARK ◽  
ADITYA POTUKUCHI ◽  
JOHN R. SCHMITT
Keyword(s):  
Upper Bound ◽  
Integral Domain ◽  
Hamming Distance ◽  
Finite Geometry ◽  
Blocking Sets ◽  
Multivariate Polynomial ◽  
Number Of Zeros ◽  
Minimum Hamming Distance ◽  
Affine Variety Codes ◽  
The Relationship

A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain ‘Condition (D)’ on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further generalized Alon–Füredi theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon–Füredi. We then discuss the relationship between Alon–Füredi and results of DeMillo–Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon–Füredi theorem and its generalization in terms of Reed–Muller-type affine variety codes is shown, which gives us the minimum Hamming distance of these codes. Then we apply the Alon–Füredi theorem to quickly recover – and sometimes strengthen – old and new results in finite geometry, including the Jamison–Brouwer–Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.

scholarly journals ON DISCRETE VALUE DISTRIBUTION OF CERTAIN COMPOSITIONS

2018 ◽  
Vol 24 (1) ◽  
pp. 34-52 ◽  
Author(s):  
Virginija Garbaliauskienė ◽  
Julija Karaliūnaitė ◽  
Renata Macaitienė
Keyword(s):  
Analytic Functions ◽  
Lower Estimate ◽  
Zeta Functions ◽  
Value Distribution ◽  
Type Condition ◽  
Space Of Analytic Functions ◽  
Number Of Zeros

In the paper, we obtain universality theorems and a lower estimate for the number of zeros for the composition F ζ (s, α; a, b) , where F is an operator in the space of analytic functions satisfying the Lipschitz type condition, and ζ (s, α; a, b) is a collection consisting of periodic and periodic Hurwitz zeta-functions.

THE ABELIAN INTEGRALS OF A ONE-PARAMETER HAMILTONIAN SYSTEM UNDER POLYNOMIAL PERTURBATIONS

2004 ◽  
Vol 14 (07) ◽  
pp. 2449-2456 ◽  
Author(s):  
TONGHUA ZHANG ◽  
WENCHENG CHEN ◽  
MAOAN HAN ◽  
HONG ZANG
Keyword(s):  
Hamiltonian System ◽  
Upper Bound ◽  
Algebraic Curve ◽  
Open Interval ◽  
Abelian Integrals ◽  
Number Of Zeros

An upper bound Z(2,n)≤[n/2]+6[(n-1)/2]+5 is derived for the number of zeros of Abelian integrals Ĩ(h)=∮γ(h)g(x,y)dx-f(x,y)dy on the open interval (0,1/6), where γ(h) is an oval lying on the algebraic curve Hλ=(1/3)x3+(1/2)x2+λy3-y2=h, f(x,y), g(x,y) are polynomials of x and y, and n= max { deg f(x,y), deg g(x,y)}. The proof exploits the expansion of [Formula: see text] near λ=0.

ON THE NUMBER OF ZEROS OF ABELIAN INTEGRAL FOR A CUBIC ISOCHRONOUS CENTER

2012 ◽  
Vol 22 (01) ◽  
pp. 1250016 ◽  
Author(s):  
KUILIN WU ◽  
YUNLIN ZHAO
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Upper Bound ◽  
Abelian Integral ◽  
Isochronous Center ◽  
Number Of Zeros ◽  
Period Annulus

In this paper, we study the number of limit cycles that bifurcate from the periodic orbits of a cubic reversible isochronous center under cubic perturbations. It is proved that in this situation the least upper bound for the number of zeros (taking into account the multiplicity) of the Abelian integral associated with the system is equal to four. Moreover, for each k = 0, 1, …, 4, there are perturbations that give rise to exactly k limit cycles bifurcating from the period annulus.

scholarly journals Some trigonometric extremal problems and duality

1991 ◽  
Vol 50 (3) ◽  
pp. 384-390 ◽  
Author(s):  
Szilárd GY. Révész
Keyword(s):  
Functional Analysis ◽  
Crucial Role ◽  
Duality Theorem ◽  
Minimax Theorem ◽  
Zeta Functions ◽  
Prime Number Theorem ◽  
Extremal Problems ◽  
Duality Results ◽  
Number Of Zeros

AbstractIn this paper we present a minimax theorem of infinite dimension. The result contains several earlier duality results for various trigonometrical extremal problems including a problem of Fejér. Also the present duality theorem plays a crucial role in the determination of the exact number of zeros of certain Beurling zeta functions, and hence leads to a considerable generalization of the classical Beurling's Prime Number Theorem. The proof used functional analysis.

THE NUMBER OF ZEROS OF ABELIAN INTEGRALS FOR A PERTURBATION OF HYPERELLIPTIC HAMILTONIAN SYSTEM WITH DEGENERATED POLYCYCLE

2013 ◽  
Vol 23 (03) ◽  
pp. 1350047 ◽  
Author(s):  
JIHUA WANG ◽  
DONGMEI XIAO ◽  
MAOAN HAN
Keyword(s):  
Asymptotic Expansion ◽  
Hamiltonian System ◽  
Upper Bound ◽  
Abelian Integral ◽  
Abelian Integrals ◽  
Exceptional Family ◽  
Level Curves ◽  
Number Of Zeros ◽  
Complete Study ◽  
Hyperbolic Saddle

In this paper, we provide a complete study of the zeros of Abelian integrals obtained by integrating the 1-form (α + βx + x2)ydx over the compact level curves of the hyperelliptic Hamiltonian [Formula: see text]. Such a family of compact level curves is bounded by a polycycle passing through a nilpotent cusp and a hyperbolic saddle of this hyperelliptic Hamiltonian system, which is not the exceptional family of ovals proposed by Gavrilov and Iliev. It is shown that the least upper bound for the number of zeros of the related hyperelliptic Abelian integral is two, and this least upper bound can be achieved for some values of parameters (α, β). This implies that the Abelian integral still has Chebyshev property for this nonexceptional family of ovals. Moreover, we derive the asymptotic expansion of Abelian integrals near a polycycle passing through a nilpotent cusp and a hyperbolic saddle in a general case.

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