### Double exponential estimate for the number of zeros of complete Abelian integrals and rational envelopes of linear ordinary differential equations with an irreducible monodromy group.

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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...

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in ${\mathbb{R}}^{n}$ and an algebraic hypersurface. The answer is polynomial in the height (the magnitude of coefficients) of the equation and the size of the curve in the space-time, with the exponent depending only on the degree and the dimension. The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial...

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in R^{n} and an affine hyperplane.
The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spanned by consecutive derivatives.
This exposition constitutes an extended abstract of a forthcoming paper: only the basic steps are outlined here, with all technical...

We study the problem of placing effective upper bounds for the number of zeroes of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of a given dimension $n$ having $m$ singular points. As a function of $n,m$, this bound turns out to be double exponential in the precise sense explained in the paper. As a corollary, we obtain a solution of the so-called restricted...

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