We prove that given a measure preserving system (X, B, µ, T 1 ,. .. , T d) with commuting, ergodic transformations T i such that T i T −1 j are ergodic for all i = j, the multicorrelation sequence a(n) = X f 0 •T n 1 f 1 •.. .•T n d f d dµ can be decomposed as a(n) = ast(n) + aer(n), where ast is a uniform limit of d-step nilsequences and aer is a nullsequence (that is, lim N −M →∞

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A recent result of Frantzikinakis in [Fra3] establishes sufficient conditions for joint ergodicity in the setting of Z-actions. We generalize this result for actions of secondcountable locally compact abelian groups. As an application, we show that, given an ergodic action (T n ) n∈F of a countable field F with characteristic zero on a probability space (X, B, µ) and a family {p 1 , . . . , p k } of F -linearly independent essentially distinct polynomials, we havewhere f i ∈ L ∞ (µ), (Φ N ) is a Følner sequence of (F, +), and the convergence takes place in L 2 (µ). We also obtain corollaries in combinatorics and topological dynamics.

<p style='text-indent:20px;'>A recent result of Frantzikinakis in [<xref ref-type="bibr" rid="b17">17</xref>] establishes sufficient conditions for joint ergodicity in the setting of <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-actions. We generalize this result for actions of second-countable locally compact abelian groups. We obtain two applications of this result. First, we show that, given an ergodic action <inline-formula><tex-math id="M2">\begin{document}$ (T_n)_{n \in F} $\end{document}</tex-math></inline-formula> of a countable field <inline-formula><tex-math id="M3">\begin{document}$ F $\end{document}</tex-math></inline-formula> with characteristic zero on a probability space <inline-formula><tex-math id="M4">\begin{document}$ (X,\mathcal{B},\mu) $\end{document}</tex-math></inline-formula> and a family <inline-formula><tex-math id="M5">\begin{document}$ \{p_1,\dots,p_k\} $\end{document}</tex-math></inline-formula> of independent polynomials, we have</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \lim\limits_{N \to \infty} \frac{1}{|\Phi_N|}\sum\limits_{n \in \Phi_N} T_{p_1(n)}f_1\cdots T_{p_k(n)}f_k\ = \ \prod\limits_{j = 1}^k \int_X f_i \ d\mu, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M6">\begin{document}$ f_i \in L^{\infty}(\mu) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ (\Phi_N) $\end{document}</tex-math></inline-formula> is a Følner sequence of <inline-formula><tex-math id="M8">\begin{document}$ (F,+) $\end{document}</tex-math></inline-formula>, and the convergence takes place in <inline-formula><tex-math id="M9">\begin{document}$ L^2(\mu) $\end{document}</tex-math></inline-formula>. This yields corollaries in combinatorics and topological dynamics. Second, we prove that a similar result holds for totally ergodic actions of suitable rings.</p>

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