  Let $\{X_i(t):\, t\in S\subset \R^d \}_{i=1,2,\ldots,n}$ be independent
copies of a stationary centered Gaussian field with almost surely smooth sample
paths. In this paper, we are interested in the conjunction probability defined
as $\PP \left( \exists t\in S: \, X_i(t) \geq u, \, \forall i=1,2,\ldots,n
\right)$ for a given threshold level $u$. As $u\rightarrow \infty$, we will
provide an asymptotic formula for the conjunction probability. This asymptotic
formula is derived from the behaviour of the volume of the set of local maximum
points. The proof relies on a result of Aza\"\i s and Wschebor describing the
shape of the excursion set of a stationary centered Gaussian field. Our result
confirms partially the validity of Euler characteristic method.
