Corners with polynomial side length

A P-polynomial corner, for P ∈ Z[z] a polynomial, is a triple of points $(x,y),\; (x+P(z),y),\; (x,y+P(z))$ for x,y,z ∈ Z. In the case where P has an integer root of multiplicity $1$, we show that if A ⊆ [N]² does not contain any nontrivial P-polynomial corners, then |A| P N²/(logloglog 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 z²-1. The main ingredients in our proof are a multidimensional quantitative concatenation result from our companion paper {KKL24a} and a novel degree-lowering argument for box norms.