  A four dimensional scalar field theory with quartic and of higher power
interactions suffers the triviality issue at the quantum level. This is due to
coupling constants that, contrary to the physical expectations, seem to grow
without a bound with energy. Since this problem concerns the high energy
domain, interaction with a quantum gravitational field may provide natural
solution to it. In this paper we address this problem considering a scalar
field theory with a general analytic potential having $\mathbb{Z}_2$ symmetry
and interacting with a quantum gravitational field. The dynamics of the latter
is governed by the cosmological constant and the Einstein-Hilbert term both
being the lowest and next-to-the lowest terms of the effective theory of
quantum gravity. Using the Vilkovisky-DeWitt method we calculate the one loop
correction to the scalar field effective action. We also derive the unique one
loop beta functions for all the scalar field couplings in the MS scheme. We
find that the leading gravitational corrections act in the direction of
asymptotic freedom. Moreover, assuming for both constants the Newton and the
cosmological to have non-zero fixed point values we find asymptotically free
Halpern-Huang potentials.
