  We calculate the two, three, four, and five-body (state independent)
effective potentials between the centers of mass (CM) of self avoiding walk
polymers by Monte-Carlo simulations. For full overlap, these coarse-grained
n-body interactions oscillate in sign as (-1)^n, and decrease in absolute
magnitude with increasing n. We find semi-quantitative agreement with a scaling
theory, and use this to discuss how the coarse-grained free energy converges
when expanded to arbitrary order in the many-body potentials. We also derive
effective {\em density dependent} 2-body potentials which exactly reproduce the
pair-correlations between the CM of the self avoiding walk polymers. The
density dependence of these pair potentials can be largely understood from the
effects of the {\em density independent} 3-body potential. Triplet correlations
between the CM of the polymers are surprisingly well, but not exactly,
described by our coarse-grained effective pair potential picture. In fact, we
demonstrate that a pair-potential cannot simultaneously reproduce the two and
three body correlations in a system with many-body interactions. However, the
deviations that do occur in our system are very small, and can be explained by
the direct influence of 3-body potentials.
