  A minimal system $(X,T)$ is topologically mildly mixing if all non-empty open
subsets $U,V$, $\{n\in \Z: U\cap T^{-n}V\neq \emptyset\}$ is an IP$^*$-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
$(X,T)$ is a topologically mildly mixing minimal system, $d\in \N$,
$p_1(n),\ldots, p_d(n)$ are integral polynomials with no $p_i$ and no $p_i-p_j$
constant, $1\le i\neq j\le d$, then for all non-empty open subsets $U , V_1,
\ldots, V_d $, $\{n\in \Z: U\cap T^{-p_1(n) }V_1\cap T^{-p_2(n)}V_2\cap \ldots
\cap T^{-p_d(n) }V_d \neq \emptyset \}$ is an IP$^*$-set. We also give the
theorem for systems under abelian group actions.

Dynamical Systems