  A $P$-polynomial corner, for $P \in \mathbb{Z}[z]$ a polynomial, is a triple
of points $(x,y),\; (x+P(z),y),\; (x,y+P(z))$ for $x,y,z \in \mathbb{Z}$. In
the case where $P$ has an integer root of multiplicity $1$, we show that if $A
\subseteq [N]^2$ does not contain any nontrivial $P$-polynomial corners, then
$$|A| \ll_P \frac{N^2}{(\log\log\log N)^c}$$ for some absolute constant $c>0$.
This simultaneously generalizes a result of Shkredov about corner-free sets and
a recent result of Peluse, Sah, and Sawhney about sets without $3$-term
arithmetic progressions of common difference $z^2-1$. The main ingredients in
our proof are a multidimensional quantitative concatenation result from our
companion paper \cite{KKL24a} and a novel degree-lowering argument for box
norms.
