一类二次可逆系统Abel积分的零点个数.PDF

35 5 () http: / /www. kustjourna.l com / V ol 35 N o5 20 10 10 Journal ofK unm ing U n iversity of Science and T echnology ( Science and T echnology) Oct 2010 do:i 10 3969 / j issn 1007- 855x 2010 05 023 Abel 黎 明 ( , 6550 11) : A rnold V I在 1977 年提出了一个弱化的 H ilbert第十六问题, 对于可积的非 H am iton 系 统, 当积分因子M (x, y) 的指数为分数时, 研 十分困难 本文利用 P icard- Fuchs方程和 R iccat i 方程, 研 了一类二次可逆系统在 n次扰动下 A b le积分零点 个数的上界问题, 得到了较小的上 界估计为 14n - 17. : 二次可逆系统; A bel积分; 线性估计 : O 175 12 : A : 1007- 855X ( 2010) 05- 0107 - 05 Num ber of Z eros of Abelian I ntegals for a C lass of uadratic R eversible System s L I M ing ( Co llege o f M athema tics and Inform a tion Sc ience, Q u jing N orm a lU n iversity, Q u jing, Y unnan 655011, China) th Abstract: In 1977, A rno ld V I put forw ard the w eak H ilbert 16 prob lem, wh ich says it is qu ite hard to study the in tegra l non- H am ilton system when the indexes o f the integra l factors o fM (x, y) are fractions B ased on the P icard- Fuchs as w ell as the R iccat i equations, this paper d iscu sses the problem o f the quadratic reversible sys- tem under polynom ial perturbat ions o f arbitrary degree n, and draw s the conc lu sion that the least upper bound of the number of zeros in the A be lian integrals is 14n- 17 K ey w ord s: quadra tic reversible system; A belian integal; linear estmi a tion 0 1900 , D H ilbert, : n P (x, y) Q (x, y ), : n n x= P (x, y), y=