2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/164620For a simple complete ideal $\wp$ of a local ring at a closed point on a smooth complex algebraic surface, we introduce an algebraic object, named Poincaré series $P_{\wp}$, that gathers in an unified way the jumping numbers and the dimensions of the vector space quotients given by consecutive multiplier ideals attached to $\wp$. This paper is devoted to prove that $P_{\wp}$ is a rational function giving an explicit expression for it.Algebraic Geometry14B05; 13H05The Poincare series of multiplier ideals of a simple complete ideal in a local ring of a smooth surfacetext