Let \(T_1, \ldots, T_d\colon X \rightarrow X\) be a family of
commuting invertible and measure preserving mappings on \((X,
\mathcal{B}, \mu)\). Let
\[
\mathcal{P} = \big(\mathcal{P}_1, \ldots, \mathcal{P}_d\big) :
\mathbb{Z}^k \rightarrow \mathbb{Z}^d
\]
be a mapping such that each \(\mathcal{P}_j\) is an integer-valued
polynomial on \(\mathbb{Z}^k\) with \(\mathcal{P}_j(0) = 0\). We
present a higher dimensional counterpart of Bourgain's pointwise
ergodic theorem along \(\mathcal{P}\). We achieve this by proving
variational estimates \(V_r\) on \(L^p(X, \mu)\) for \(p > 1\) and
\(r > \max\{p, p/(p-1)\}\) for an averaging operator
\[
M_N f(x) = \frac{1}{N^k} \sum_{y_1 = 1}^ N \cdots \sum_{y_k = 1}^N
f\big(T_1^{\mathcal{P}_1(n)} \cdots T_d^{\mathcal{P}_d(n)} x\big).
\]
Moreover, we obtain the estimates which are uniform in the
coefficients of \(\mathcal{P}\).
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