  If one-electron observables of a many-electron system are of interest, a
many-electron dynamics can be represented exactly by a one-electron dynamics
with effective potentials. The formalism for this reduction is provided by the
Exact Electron Factorization (EEF). We study the time-dependent features of the
EEF effective potentials for a model of an atom ionized by an ultrastrong and
ultrashort laser pulse, with the aim of understanding what is needed to develop
computationally feasible approximations. It is found that the simplest
approximation, the so-called time-independent conditional amplitude (TICA)
approximation, is complementary to single-active electron (SAE) approaches as
it reproduced the exact dynamics well for high photon frequencies of the laser
field or large Keldysh parameter. For relatively low frequencies of the laser
field or for smaller Keldysh parameters, we find that excited state dynamics in
the core region of the atom leads to a time-dependent ionization barrier in the
EEF potential. The time-dependence of the barrier needs to be described
accurately to correctly model many-electron effects, and we conclude that a
multi-state extension of the TICA approximation is a possible route how this
can be achieved. In general, our study sheds a different light on one-electron
pictures of strong-field ionization and shows that many-electron effects for
such processes may be included by solving a one-electron Schr\"odinger
equation, provided the core dynamics can be modeled successfully.
