  In 2012, we applied a three-dimensional formulation to kinematic measurements
of the Galactic thick disk and derived a surprisingly low dark matter density
at the solar position. This result was challenged by Bovy & Tremaine (2012,
ApJ, 756, 89), who claimed that the observational data are consistent with the
expected dark matter density if a one-dimensional approach is adopted. We
analyze the assumption at the bases of their formulation and their claim that
this returns a lower limit for the local dark matter density, which is accurate
within 20%. We find that the validity of their formulation depends on the
underlying mass distribution. We therefore analyze the predictions that their
hypothesis casts on the radial gradient of the azimuthal velocity dV/dR and
compare it with observational data as a testbed for the validity of their
formulation. We find that their hypothesis requires too steep a profile of
dV(Z)/dR, which is inconsistent with the observational data both in the Milky
Way and in external galaxies. As a consequence, their results are biased and
largely overestimate the mass density. Dynamical simulations also show that,
contrary to their claims, low values of dV/dR are compatible with a Milky Way
like potential with radially constant circular velocity. We nevertheless
confirm that, according to their criticism, our assumption dV/dR=0 is only an
approximation. If this hypothesis is released, and the available information
about dV/dR in the thick disk is used, the resulting local dark matter density
increases by a tiny amount, from 0+-1 to 2+-3 mM_sun pc^(-3), with an upper
limit of ~3.5 mM_sun pc^(-3). Hence, this approximation has negligible
influence on our results. Our analysis shows that their criticism is not a
viable explanation for the inferred lack of dark matter at the solar position
detected by us. More studies are required to understand these unexpected
results.
