The sharp bound on the number of zeros of Abelian integral for a perturbation of hyper-elliptic Hamiltonian system

Abstract

Listen

Translate article

In this paper, we study the number of zeros of the Abelian integral ∮ Г h (a 0 + a 1 x + a 2 x 2 + a 3 x 3 ) ydx de ned on the compact level curves Г h of the quintic hyper-elliptic Hamiltonian: H ( x , y ) = (1/2) y 2 + (9/2) x 2 + 5 x 3 + (7/4) x 4 + (1/5) x 5 . It is proved that 3 is an upper bound of the number of zeros of the above Abelian integral and 3 zeros can be reached. The proof relies on a Chebyshev criterion (Grau et al. (2011)), some techniques in mathematical mechanization and the asymptotic expansions of Abelian integral.