2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71862We elaborate notions of integration over the space of arcs factorized by the natural $C^*$-action and over the space of non-parametrized arcs (branches). There are offered two motivic versions of the zeta function of the classical monodromy transformation of a germ of an analytic function on a smooth space. We indicate a direct formula which connects the naive motivic zeta function of J. Denef and F. Loeser with the classical monodromy zeta function.Algebraic Geometry14B05, 32S05, 58K10text