  Szemer\'edi's Theorem states that a set of integers with positive upper
density contains arbitrarily long arithmetic progressions. Bergelson and
Leibman generalized this, showing that sets of integers with positive upper
density contain arbitrarily long polynomial configurations; Szemer\'edi's
Theorem corresponds to the linear case of the polynomial theorem. We focus on
the case farthest from the linear case, that of rationally independent
polynomials. We derive results in ergodic theory and in combinatorics for
rationally independent polynomials, showing that their behavior differs sharply
from the general situation.
