For $ellgeq 2$ and $hin mathbb{Z}[x_1,dots,x_{ell}]$ of degree $kgeq 2$, we show that every set $Asubseteq {1,2,dots,N}$ 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 $ellgeq 3$, 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.
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