2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/209909Starting from certain rational varieties blown-up from (P^1)^N, we construct a tropical, i.e., subtraction-free birational, representation of Weyl groups as a group of pseudo isomorphisms of the varieties. Furthermore, we develop an algebro-geometric framework of tau-functions as defining functions of exceptional divisors on the varieties. In the case where the corresponding root system is of affine type, our construction yields a class of (higher order) q-difference Painleve equations and its algebraic degree grows quadratically.18 pages, 3 figures; corrected mainly in Sect.4 and 5Algebraic Geometry14E07; 14L30; 20F55; 34M55; 37K10; 39A13text