2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/136136Let $k$ be a finitely generated field, let $X$ be an algebraic variety and $G$ a linear algebraic group, both defined over $k$. Suppose $G$ acts on $X$ and every element of a Zariski-dense semigroup $Γ\subset G(k)$ has a rational fixed point in $X(k)$. We then deduce, under some mild technical assumptions, the existence of a rational map $G\to X$, defined over $k$, sending each element $g\in G$ to a fixed point for $g$. The proof makes use of a recent result of Ferretti and Zannier on diophantine equations involving linear recurrences. As a by-product of the proof, we obtain a version of the classical Hilbert Irreducibility Theorem valid for linear algebraic groups.35 pages, Plain Tex. A gap in the previous proof of Theorem 1.2 overcome, plus minor changes. Thanks to J. Bernik and the refereeNumber TheoryAlgebraic Geometry11G35; 12E25; 11E99text