Geometry of conjunction set of smooth stationary Gaussian fields.

We investigate the conjunction probability that at a same point the values of the Gaussian fields exceed the given threshold. This problem was studied by Double-sum method or Euler characteristic method. In this paper, using a recent result of Aza\"i s and Wschebor describing the shape of the excursion set, we give the explicit values of the generalized Pickands constants and compare them with the predictions by Euler characteristic method. Our results give a partial validity of Euler characteristic method.