Abelian integrals for cubic vector fields (Q1570418)

The authors prove that the lowest upper bound of the number of the isolated zeros to the Abelian integral \[ I(h)= \oint_{\Gamma_h}(\alpha+ \beta x+\gamma x^2)y dx \] is two for \(h\in (-1/12,0)\), where \(\Gamma_h\) is the compact component of \(H(x, y)= (1/2)y^2+ (1/3) x^3+ (1/4)x^4= h,\alpha, \beta,\gamma\) are arbitrary constants.