Continuité des racines d’après Rabinoff

The content of this paper is a generalization of a theorem by Joseph Rabinoff: if $\mathscr{P}$ is a finite family of pointed and rational polyhedra in $N_\mathbb{R}$ such that there exists a fan in $N_\mathbb{R}$ that contains all the recession cones of the polyhedra of $\mathscr{P}$, if $k$ is a complete non-archimedean field, if $S$ is a connected and regular $k$-analytic space (in the sense of Berkovich) and $Y$ is a closed $k$-analytic subset of $U_{\mathscr{P}} \times _k S$ which is relative complete intersection and contained in the relative interior of $U_{\mathscr{P}} \times _k S$ over $S$, then the quasifiniteness of $\pi : Y \rightarrow S$ implies its flatness and finiteness; moreover, all the finite fibers of $\pi $ have the same length. This namely gives a analytic justification to the concept of stable intersection used in the theory of tropical intersection.