  For $\ell\geq 2$ and $h\in \mathbb{Z}[x_1,\dots,x_{\ell}]$ of degree $k\geq
2$, we show that every set $A\subseteq \{1,2,\dots,N\}$ lacking nonzero
differences in $h(\mathbb{Z}^{\ell})$ satisfies $|A|\ll_h Ne^{-c(\log
N)^{\mu}}$, where $c=c(h)>0$, $\mu=[(k-1)^2+1]^{-1}$ if $\ell=2$, and $\mu=1/2$
if $\ell\geq 3$, provided $h(\mathbb{Z}^{\ell})$ contains a multiple of every
natural number and $h$ satisfies certain nonsingularity conditions. We also
explore these conditions in detail, drawing on a variety of tools from
algebraic geometry.
