  While the standard and most popular explanation for the flatness of galaxy
rotation curves is dark matter, one cannot at this stage rule out an
explanation based on a modified law of gravitation, which agrees with Newtonian
gravitation on the scale of the solar system, but differs from it on larger
length scales. Examples include Modfied Newtonian Dynamics [MOND] and
Scalar-Tensor-Vector Gravity [STVG]. Here we report on a fourth order
modification of the Poisson equation which yields the same Yukawa type
modification of Newtonian gravity as STVG, and which can explain flat galaxy
rotation curves for a large sample of galaxies, once specific values for two
parameters have been chosen. We speculate on two possible origins for this
modified Poisson equation: first, a possible fourth order modification of
general relativity, and second, quadrupole gravitational polarization induced
on a galaxy because of the pull of neighbouring galaxies.
