  This paper aims at understanding if the normalization of the stellar initial
mass function (IMF) of massive early-type galaxies (ETGs) varies with cosmic
time and/or with mean stellar mass density Sigma (M*/2\pi Re^2). For this
purpose we collected a sample of 18 dense (Sigma>2500 M_sun/pc^2) ETGs at
1.2<z<1.6 with available velocity dispersion sigma_e. We have constrained their
mass-normalization by comparing their true stellar masses (M_true) derived
through virial theorem, hence IMF independent, with those inferred through the
fit of the photometry assuming a reference IMF (M_ref). Adopting the virial
estimator as proxy of the true stellar mass, we have assumed for these ETGs
zero dark matter (DM). However, dynamical models and numerical simulations of
galaxy evolution have shown that the DM fraction within Re in dense high-z ETGs
is negligible. We have considered the possible bias of virial theorem in
recovering the total masses and have shown that for dense ETGs the virial
masses are in agreement with those derived through more sophisticated dynamical
models. The variation of the parameter Gamma = M_true/M_ref with sigma_e shows
that, on average, dense ETGs at <z> = 1.4 follow the same IMF-sigma_e trend of
typical local ETGs, but with a lower mass-normalization. Nonetheless, once the
IMF-sigma_e trend we have found for high-z dense ETGs is compared with that of
local ETGs with similar Sigma and sigma_e, they turn out to be consistent. The
similarity between the IMF-sigma_e trends of dense high-z and low-z ETGs over 9
Gyr of evolution and their lower mass-normalization with respect to the mean
value of local ETGs suggest that, independently on formation redshift, the
physical conditions characterizing the formation of a dense spheroid lead to a
mass spectrum of new formed stars with an higher ratio of high- to low-mass
stars with respect to the IMF of normal local ETGs.
