  We prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in
(1,\infty)$, for certain multi-parameter polynomial ergodic averages by
establishing the corresponding multi-parameter maximal and oscillation
inequalities. Our result, in particular, gives an affirmative answer to a
multi-parameter variant of the Bellow-Furstenberg problem. This paper is also
the first systematic treatment of multi-parameter oscillation semi-norms which
allows an efficient handling of multi-parameter pointwise convergence problems
with arithmetic features. The methods of proof of our main result develop
estimates for multi-parameter exponential sums, as well as introduce new ideas
from the so-called multi-parameter circle method in the context of the geometry
of backwards Newton diagrams that are dictated by the shape of the polynomials
defining our ergodic averages.
