On the number of zeros of Abelian integrals for a kind of quartic Hamiltonians

Juanjuan Wu; Yongkang Zhang; Cuiping Li

doi:10.1016/j.amc.2013.11.092

On the number of zeros of Abelian integrals for a kind of quartic Hamiltonians

Juanjuan Wu^*
, Yongkang Zhang
, Cuiping Li

^*Corresponding author for this work

School of Mathematical Sciences

Beihang University
Tianjin Normal University

Research output: Contribution to journal › Article › peer-review

15 Scopus citations

Abstract

An explicit upper bound B(n) is derived for the number of zeros of Abelian integrals I(h)=^Γhg(x,y)dx-f(x,y)dy on the open interval (0,1/4), where ^Γh is an oval lying on the algebraic curve H(x,y)=^x2+^y2-^x4+^ax2 ^y2+^y4 with a>-2,f(x,y) and g(x,y) are polynomials in x and y of degrees not exceeding n. Assume I(h) not vanish identically, then B(n)≤3n-14+12n-34+23.

Original language	English
Pages (from-to)	329-335
Number of pages	7
Journal	Applied Mathematics and Computation
Volume	228
DOIs	https://doi.org/10.1016/j.amc.2013.11.092
State	Published - 1 Feb 2014

Keywords

Abelian integrals
Poincare bifurcation
Quartic Hamiltonian
Weakened Hilbert 16th problem

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10.1016/j.amc.2013.11.092

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abstract = "An explicit upper bound B(n) is derived for the number of zeros of Abelian integrals I(h)=Γhg(x,y)dx-f(x,y)dy on the open interval (0,1/4), where Γh is an oval lying on the algebraic curve H(x,y)=x2+y2-x4+ax2 y2+y4 with a>-2,f(x,y) and g(x,y) are polynomials in x and y of degrees not exceeding n. Assume I(h) not vanish identically, then B(n)≤3n-14+12n-34+23.",

keywords = "Abelian integrals, Poincare bifurcation, Quartic Hamiltonian, Weakened Hilbert 16th problem",

author = "Juanjuan Wu and Yongkang Zhang and Cuiping Li",

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doi = "10.1016/j.amc.2013.11.092",

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T1 - On the number of zeros of Abelian integrals for a kind of quartic Hamiltonians

AU - Wu, Juanjuan

AU - Zhang, Yongkang

AU - Li, Cuiping

PY - 2014/2/1

Y1 - 2014/2/1

N2 - An explicit upper bound B(n) is derived for the number of zeros of Abelian integrals I(h)=Γhg(x,y)dx-f(x,y)dy on the open interval (0,1/4), where Γh is an oval lying on the algebraic curve H(x,y)=x2+y2-x4+ax2 y2+y4 with a>-2,f(x,y) and g(x,y) are polynomials in x and y of degrees not exceeding n. Assume I(h) not vanish identically, then B(n)≤3n-14+12n-34+23.

AB - An explicit upper bound B(n) is derived for the number of zeros of Abelian integrals I(h)=Γhg(x,y)dx-f(x,y)dy on the open interval (0,1/4), where Γh is an oval lying on the algebraic curve H(x,y)=x2+y2-x4+ax2 y2+y4 with a>-2,f(x,y) and g(x,y) are polynomials in x and y of degrees not exceeding n. Assume I(h) not vanish identically, then B(n)≤3n-14+12n-34+23.

KW - Abelian integrals

KW - Poincare bifurcation

KW - Quartic Hamiltonian

KW - Weakened Hilbert 16th problem

UR - https://www.scopus.com/pages/publications/84891081807

U2 - 10.1016/j.amc.2013.11.092

DO - 10.1016/j.amc.2013.11.092

M3 - 文章

AN - SCOPUS:84891081807

SN - 0096-3003

VL - 228

SP - 329

EP - 335

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

ER -

On the number of zeros of Abelian integrals for a kind of quartic Hamiltonians

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Keywords

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