  We study isotropic Gaussian random fields on the high-dimensional sphere with
an added deterministic linear term, also known as mixed p-spin Hamiltonians
with external field. We prove that if the external field is sufficiently
strong, then the resulting function has trivial geometry, that is only two
critical points. This contrasts with the situation of no or weak external field
where these functions typically have an exponential number of critical points.
We give an explicit threshold $h_c$ for the magnitude of the external
fieldnecessary for trivialization and conjecture $h_c$ to be sharp. The
Kac-Rice formula is our main tool. Our work extends [Fyo15], which identified
the trivial regime for the special case of pure p-spin Hamiltonians with random
external field.
