  A nonlinear version of Roth's theorem states that dense sets of integers
contain configurations of the form $x$, $x+d$, $x+d^2$. We obtain a
multidimensional version of this result, which can be regarded as a first step
towards effectivising those cases of the multidimensional polynomial
Szemer\'edi theorem involving polynomials with distinct degrees. In addition,
we prove an effective ``popular'' version of this result, showing that every
dense set has some non-zero $d$ such that the number of configurations with
difference parameter $d$ is almost optimal. Perhaps surprisingly, the
quantitative dependence in this result is exponential, compared to the
tower-type bounds encountered in the popular linear Roth theorem.
