  Using high-resolution echelle spectra obtained with Magellan/MIKE, we present
a chemical abundance analysis of both stars in the planet-hosting wide binary
system HD20782 + HD20781. Both stars are G dwarfs, and presumably coeval,
forming in the same molecular cloud. Therefore we expect that they should
possess the same bulk metallicities. Furthermore, both stars also host giant
planets on eccentric orbits with pericenters $\lesssim 0.2\,$ AU. We
investigate if planets with such orbits could lead to the host stars ingesting
material, which in turn may leave similar chemical imprints in their
atmospheric abundances. We derived abundances of 15 elements spanning a range
of condensation temperatures ($T_{C}\approx 40-1660\,$ K). The two stars are
found to have a mean element-to-element abundance difference of $0.04\pm0.07\,$
dex, which is consistent with both stars having identical bulk metallicities.
In addition, for both stars, the refractory elements ($T_{C} > 900\,$ K)
exhibit a positive correlation between abundance (relative to solar) and
$T_{C}$, with similar slopes of $\approx$ $1\times10^{-4}\,$ dex K$^{-1}$. The
measured positive correlations are not perfect; both stars exhibit a scatter of
$\approx$ $5\times10^{-5}\,$ dex K$^{-1}$ about the mean trend, and certain
elements (Na, Al, Sc) are similarly deviant in both stars. These findings are
discussed in the context of models for giant planet migration that predict the
accretion of H-depleted rocky material by the host star. We show that a simple
simulation of a solar-type star accreting material with Earth-like composition
predicts a positive---but imperfect---correlation between refractory elemental
abundances and $T_{C}$. Our measured slopes for HD 20782/81 are consistent with
what is predicted for the ingestion of 10--20 Earths by both stars.
