  The diffusion-controlled reaction $kA\rightarrow\emptyset$ is known to be
strongly dependent on fluctuations in dimensions $d\le d_c=2/(k-1)$. We develop
a field theoretic renormalization group approach to this system which allows
explicit calculation of the observables as expansions in $\epsilon^{1/(k-1)}$,
where $\epsilon=d_c-d$. For the density it is found that, asymptotically,
$n\sim A_k t^{-d/2}$. The decay exponent is exact to all orders in $\epsilon$,
and the amplitude $A_k$ is universal, and is calculated to second order in
$\epsilon^{1/(k-1)}$ for $k=2,3$. The correlation function is calculated to
first order, along with a long wavelength expansion for the second order term.
For $d=d_c$ we find $n \sim A_k (\ln t/t)^{1/(k-1)}$ with an exact expression
for $A_k$. The formalism can be immediately generalized to the reaction
$kA\rightarrow\ell A$, $\ell < k$, with the consequence that the density
exponent is the same, but the amplitude is modified.
