  To an ideal in $\mathbb{C}[x,y]$ one can associate a topological zeta
function. This is an extension of the topological zeta function associated to
one polynomial. But in this case we use a principalization of the ideal instead
of an embedded resolution of the curve.
  In this paper we will study two questions about the poles of this zeta
function. First, we will give a criterion to determine whether or not a
candidate pole is a pole. It turns out that we can know this immediately by
looking at the intersection diagram of the principalization, together with the
numerical data of the exceptional curves. Afterwards we will completely
describe the set of rational numbers that can occur as poles of a topological
zeta function associated to an ideal in dimension two. The same results are
valid for related zeta functions, as for instance the motivic zeta function.
