Finding the mean of the total number N_tot of stationary points for N-dimensional random Gaussian landscapes can be reduced to averaging the absolute value of characteristic polynomial of the corresponding Hessian. First such a reduction is illustrated for a class of models describing energy landscapes of elastic manifolds in random environment, and a general method of attacking the problem analytically is suggested. Then the exact solution to the problem (Y.V. Fyodorov,  Phys. Rev. Lett. 93, 149901(E) ( 2004) )  for a class of landscapes corresponding to the simplest, yet nontrivial ``toy model'' with N degrees of freedom is described. For N \gg 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: N_tot \sim e^{N{\Sigma }}, {\Sigma }(\mu <\mu _c)>0 to the phase with vanishing complexity: {\Sigma }(\mu >\mu _c) = 0. This is interpreted as a transition to a glass-like state of the matter.

PACS numbers: 05.40.--a, 75.10.Nr