Reasonable bounds for Restricted Polynomial Progressions in ${\mathbb{F}}_p^n$

Given an integer $k\ge 2$ and a prime $p$ large enough depending on $k$, we show that, if $A\subseteq {\mathbb{F}}_p^n$ does not contain a restricted polynomial progression of the form \begin{align*} x, x+\begin{pmatrix} y_1 \ y_2 \ \vdots \ y_n \end{pmatrix}, x+\begin{pmatrix} y_1^2 \ y_2^2 \ \vdots \ y_n^2 \end{pmatrix}, \ldots, x+\begin{pmatrix} y_1^{k-1} \ y_2^{k-1} \ \vdots \ y_n^{k-1} \end{pmatrix}, \end{align*} where $x\in {\mathbb{F}}_p^n$ and $y\in \{0,2,3,\dots ,M_k{\}}^n\mathrm{\backslash }\{\vec{0}\}$, for some fixed $M_k$, then $A$ must have density bounded above by $\frac{C}{({\log }^{(k^{O(k)})}n{)}^c}$, where $C,c>0$ only depend on $k$ and $p$. For $k\ge 4$, the best previously known bound came from the density Hales–Jewett theorem and was of inverse Ackermann type. The proof uses a recent inverse theorem for noise stability of Bhangale, Khot, Liu and Minzer, which arose in the study of constraint satisfaction problems. The main new ingredient is a proof that the underlying distribution counting restricted polynomial progressions does not admit a nontrivial abelian embedding.