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| 摘要: |
| 得到了Abel积分$I(h)=\oint_{\Gamma_h}g(x,y)dx-f(x,y)dy$在开区间$(0,\frac{1}{4})$上零点个数$B(n)\leq3\Big[\frac{n-1}{4}\Big]$,
其中$\Gamma_h$是代数曲线$H(x,y)=x^4+y^4-x^8=h,\ h\in(0,\frac{1}{4})$所定义的卵形线,
$f(x,y)=\sum\limits_{1\leq4i+4j+1\leq n}a_{ij}x^{4i+1}y^{4j}$和
$g(x,y)=\sum\limits_{1\leq4i+4j+1\leq n}b_{ij}x^{4i}y^{4j+1}$是$x$和$y$的次数不超过$n$的多项式. |
| 关键词: Hamilton系统 幂零奇点 Abel积分 Picard-Fuchs方程 |
| DOI： |
| 分类号:O175.12 |
| 基金项目:国家自然科学基金项目（面上项目，重点项目，重大项目） |
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| On the number of zeros for Abel integrals of Hamilton system of seven degree with nilpotent singularities |
|
MA Huilong,YANG Jihua
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| Abstract: |
| An upper bound $B(n)\leq3[\frac{n-1}{4}]$ is derived for the number of zeros of Abel integrals
$I(h)=\oint_{\Gamma_h}g(x,y)dy-f(x,y)dx$ on the open interval $(0,\frac{1}{4})$, where $\Gamma_h$ is an oval lying on
the algebraic curve $H(x,y)=x^4+y^4-x^8=h,\ h\in(0,\frac{1}{4})$, $f(x,y)=\sum\limits_{1\leq4i+4j+1\leq n}x^{4i+1}y^{4j}$ and
$g(x,y)=\sum\limits_{1\leq4i+4j+1\leq n}x^{4i}y^{4j+1}$ are polynomials of $x$ and $y$ of degrees not exceeding $n$. |
| Key words: Hamiltonian system nilpotent singularity Abelian integral Picard-Fuchs equation |
