### Simple exponential estimate for the number of real zeros of complete abelian integrals

We show that for a generic polynomial $H=H(x,y)$ and an arbitrary differential 1-form $\omega =P(x,y)\phantom{\rule{0.166667em}{0ex}}dx+Q(x,y)\phantom{\rule{0.166667em}{0ex}}dy$ with polynomial coefficients of degree $\le d$, the number of ovals of the foliation $H=\mathrm{const}$, which yield the zero value of the complete Abelian integral $I\left(t\right)={\oint}_{H=t}\omega $, grows at most as $exp{O}_{H}\left(d\right)$ as $d\to \infty $, where ${O}_{H}\left(d\right)$ depends only on $H$. The main result of the paper is derived from the following more general theorem on bounds for isolated zeros occurring in polynomial envelopes of linear differential equations. Let ${f}_{1}\left(t\right),\cdots ,{f}_{n}\left(t\right)$, $t\in K\u22d0\mathbb{R}$, be a fundamental system of real solutions...