The identification of new rare signals in data, the detection of a sudden
change in a trend, and the selection of competing models, are among the most
challenging problems in statistical practice. These challenges can be tackled
using a test of hypothesis where a nuisance parameter is present only under the
alternative, and a computationally efficient solution can be obtained by the
"Testing One Hypothesis Multiple times" (TOHM) method. In the one-dimensional
setting, a fine discretization of the space of the non-identifiable parameter
is specified, and a global p-value is obtained by approximating the
distribution of the supremum of the resulting stochastic process. In this
paper, we propose a computationally efficient inferential tool to perform TOHM
in the multidimensional setting. Here, the approximations of interest typically
involve the expected Euler Characteristics (EC) of the excursion set of the
underlying random field. We introduce a simple algorithm to compute the EC in
multiple dimensions and for arbitrary large significance levels. This leads to
an highly generalizable computational tool to perform inference under
non-standard regularity conditions.