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We examine multiple ergodic averages of commuting transformations withpolynomial iterates in which the polynomials may be pairwise dependent. Inparticular, we show that such averages are controlled by the Gowers-Host-Kraseminorms whenever the system satisfies some mild ergodicity assumptions.Combining this result with the general criteria for joint ergodicityestablished in our earlier work, we determine a necessary and sufficientcondition under which such averages are jointly ergodic, in the sense that theyconverge in the mean to the product of integrals, or weakly jointly ergodic, inthat they converge to the product of conditional expectations. As a corollary,we deduce a special case of a conjecture by Donoso, Koutsogiannis, and Sun in astronger form.

Seminorm control for ergodic averages with commuting transformations and pairwise dependent polynomial iterates