[Abridged] We present the evolution of the stellar mass function (SMF) of galaxies from z=4.0 to z=1.3 measured from a sample constructed from the deep NIR MUSYC, the FIRES, and the GOODS-CDFS surveys, all having very high-quality optical to mid-infrared data. This sample, unique in that it combines data from surveys with a large range of depths and areas in a self-consistent way, allowed us to 1) minimize the uncertainty due to cosmic variance and empirically quantify its contribution to the total error budget; 2) simultaneously probe the high-mass end and the low-mass end (down to ~0.05 times the characteristic stellar mass) of the SMF with good statistics; and 3) empirically derive the redshift-dependent completeness limits in stellar mass. We provide, for the first time, a comprehensive analysis of random and systematic uncertainties affecting the derived SMFs. We find that the mass density evolves by a factor of ~17(+7,-10) since z=4.0, mostly driven by a change in the normalization Phi*. If only random errors are taken into account, we find evidence for mass-dependent evolution, with the low-mass end evolving more rapidly than the high-mass end. However, we show that this result is no longer robust when systematic uncertainties due to the SED-modeling assumptions are taken into account. Taking our results at face value, we find that they are in conflict with semi-analytic models of galaxy formation. The models predict SMFs that are in general too steep, with too many low-mass galaxies and too few high-mass galaxies. The discrepancy at the high-mass end is susceptible to uncertainties in the models and the data, but the discrepancy at the low-mass end may be more difficult to explain.
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