We show that for a generic polynomial $H=H(x,y)$ and an arbitrary differential 1-form $\omega =P(x,y)dx+Q(x,y)dy$ 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 $f_1\left(t\right),\cdots ,f_n\left(t\right)$, $t\in K⋐ℝ$, be a fundamental system of real solutions...

For a holomorphic function on a complex manifold, we show that the vanishing cohomology of lower degree at a point is determined by that for the points near it, using the perversity of the vanishing cycle complex. We calculate this order of vanishing explicitly in the case the hypersurface has simple normal crossings outside the point. We also give some applications to the size of Jordan blocks for monodromy.

Soit $X\to S$ un morphisme propre d’un $\mathbf{C}$-schéma intègre dans un germe de courbe algébrique lisse sur $\mathbf{C}$. On construit une structure de Hodge mixte sur les cohomologies évanescentes en résolvant les complexes évanescents $R{\psi }_X$ et $R{\varphi }_X$ par des complexes de Hodge mixtes cohomologiques. Ceci donne une majoration du niveau d’unipotence de l’action de la monodromie.

On étudie le comportement des faisceaux $l$-adiques entiers sur les schémas de type fini sur un corps local par les six opérations et le foncteur des cycles proches.