A refined saturation theorem for polynomials and applications

For a dynamical system $(X,T)$, $d\in \mathbb{N}$ and distinct non-constant integral polynomials $p_1,\dots ,p_d$ vanishing at $0$, the notion of regionally proximal relation along $C=\{p_1,\dots ,p_d\}$ (denoted by $R{P}_C^{[d]}(X,T)$) is introduced. It turns out that for a minimal system, $R{P}_C^{[d]}(X,T)=\mathrm{\Delta }$ implies that $X$ is an almost one-to-one extension of $X_k$ for some $k\in \mathbb{N}$ only depending on a set of finite polynomials associated with $C$ and has zero entropy, where $X_k$ is the maximal $k$-step pro-nilfactor of $X$. Particularly, when $C$ is a collection of linear polynomials, it is proved that $R{P}_C^{[d]}(X,T)=\mathrm{\Delta }$ implies $(X,T)$ is a $d$-step pro-nilsystem, which answers negatively a conjecture in \cite{5p}. The results are obtained by proving a refined saturation theorem for polynomials.