On the Chebyshev Property of a Class of Hyperelliptic Abelian Integrals

This paper aims to demonstrate the Chebyshev property of the linear space \(V=\{\sum _{i=0}^{2}\alpha _i\oint _{\Gamma _h}x^{2i}y\textrm{d}x:\alpha _0,\alpha _1,\alpha _2\in \mathbb {R},\,h\in \Sigma \}\) $V=\{{\sum }_{i=0}^2{\alpha }_i{\oint }_{{\mathrm{\Gamma }}_h}{x}^{2i}y\text{d}x:{\alpha }_0,{\alpha }_1,{\alpha }_2\in \mathbb{R},h\in \mathrm{\Sigma }\}$ (which is equivalent to that every function of V has at most 2 zeros, counted with multiplicity), with three hyperelliptic Abelian integrals \(\oint _{\Gamma _h}x^{2i}y\textrm{d}x \,(i=0,1,2)\) ${\oint }_{{\mathrm{\Gamma }}_h}{x}^{2i}y\text{d}x(i=0,1,2)$ as generators, where \(\Gamma _h\) ${\mathrm{\Gamma }}_h$ is an oval determined by \(H(x,y)=\frac{y^2}{2}+\Psi (x)=h\) $H(x,y)=\frac{y^2}{2}+\mathrm{\Psi }(x)=h$ , and \(\Psi (x)\) $\mathrm{\Psi }(x)$ is an even polynomial of indefinite degree with real non-Morse critical points. As an application, we can obtain the exact upper bound for the number of zeros of a class of hyperelliptic Abelian integrals related to some planar polynomial Hamiltonian systems with two cusps and a nilpotent center.