  We obtain a polynomial upper bound in the finite-field version of the
multidimensional polynomial Szemer\'{e}di theorem for distinct-degree
polynomials. That is, if $P_1, ..., P_t$ are nonconstant integer polynomials of
distinct degrees and $v_1, ..., v_t$ are nonzero vectors in $\mathbb{F}_p^D$,
we show that each subset of $\mathbb{F}_p^D$ lacking a nontrivial configuration
of the form
  $$ x, x + v_1 P_1(y), ..., x + v_t P_t(y)$$ has at most $O(p^{D-c})$
elements. In doing so, we apply the notion of Gowers norms along a vector
adapted from ergodic theory, which extends the classical concept of Gowers
norms on finite abelian groups.
