Topological mild mixing of all orders along polynomials

A minimal system \begin{document}$ (X,T) $\end{document} is topologically mildly mixing if for all non-empty open subsets \begin{document}$ U,V $\end{document} , \begin{document}$ \{n\in {\mathbb Z}: U\cap T^{-n}V\neq \emptyset\} $\end{document} is an IP \begin{document}$ ^* $\end{document} -set. In this paper we show that if a minimal system is topologically mildly mixing, then it is mild mixing of all orders along polynomials. That is, suppose that \begin{document}$ (X,T) $\end{document} is a topologically mildly mixing minimal system, \begin{document}$ d\in {\mathbb N} $\end{document} , \begin{document}$ p_1(n),\ldots, p_d(n) $\end{document} are integral polynomials with no \begin{document}$ p_i $\end{document} and no \begin{document}$ p_i-p_j $\end{document} constant, \begin{document}$ 1\le i\neq j\le d $\end{document} . Then for all non-empty open subsets \begin{document}$ U , V_1, \ldots, V_d $\end{document} , \begin{document}$ \{n\in {\mathbb Z}: U\cap T^{-p_1(n) }V_1\cap T^{-p_2(n)}V_2\cap \ldots \cap T^{-p_d(n) }V_d \neq \emptyset \} $\end{document} is an IP \begin{document}$ ^* $\end{document} -set. We also give the corresponding theorem for systems under abelian group actions.