  Finding the mean of the total number N_{tot} of critical points for
N-dimensional random energy landscapes is reduced to averaging the absolute
value of characteristic polynomial of the corresponding Hessian. For any finite
N we provide the exact solution to the problem for a class of landscapes
corresponding to the "toy model" of manifolds in random environment. For N >>1
our asymptotic analysis reveals a phase transition at some critical value \mu_c
of a control parameter \mu from a phase with finite landscape complexity to the
phase with vanishing complexity. The same value of the control parameter is
known to correspond to an onset of glassy behaviour at zero temperature.
Finally, we discuss a method of dealing with the modulus of the spectral
determinant applicable to a broad class of problems.
