Decomposition of multicorrelation sequences and joint ergodicity

We show that, under finitely many ergodicity assumptions, any multicorrelation sequence defined by invertible measure preserving $\mathbb{Z}^d$-actions with multivariable integer polynomial iterates is the sum of a nilsequence and null sequence, extending recent result second author. To this end, we develop new seminorm bound estimate for multiple averages improving results in previous work first, third fourth authors. also use approach to obtain criteria joint on $\mathbb{Z}^{d}$-systems.