一类具有尖点环的三次Hamilton向量场的Abel积分

Abelian Integrals for Cubic Hamiltonian System with a Cuspidal Loop

文进一步完善了文 [6]的工作 ,证明了 Abel积分I(h) =∮Γh(α +βx +γx2 ) ydx的零点个数上界 B(3)满足不等式 4≤B(3)≤ 6,这里 Γh是代数曲线 H(x,y) =12 y2 +13x3 +14 x4=h的连通闭分支 ,h∈ (- 1 / 1 2 ,0 )∪ (0 ,+∞ )

It is proved that in this paper that the lowest upper bound \$B(3)\$ of the numb er of zeros of Abelian integrals \$I(h)=∮\-\{Γ\-h\}(α+βx+γx\+2)y d x\$ sa tisfies \$4≤B(3)≤6,\$ where \$Γ\-h\$ is the compact component of \$H(x,y)=y\+ 2/2+x\+3/3+x\+4/4=h, α,β,γ\$ are arbitrary real constants.