Seminorms for multiple averages along polynomials and applications to joint ergodicity

Exploiting the recent work of Tao and Ziegler on a concatenation theorem factors, we find explicit characteristic factors for multiple averages along polynomials systems with commuting transformations, use them to study criteria joint ergodicity sequences form $(T^{p_{1,j}(n)}_{1}\cdot\ldots\cdot T^{p_{d,j}(n)}_{d})_{n\in\mathbb{Z}},$ $1\leq j\leq k$, where $T_{1},\dots,T_{d}$ are measure preserving transformations probability space $p_{i,j}$ integer polynomials. To be more precise, provide sufficient condition such jointly ergodic, giving also characterization $(T^{p(n)}_{i})_{n\in\mathbb{Z}}, 1\leq i\leq d$ answering question due Bergelson.