Polynomial corners in finite fields beyond the distinct-degree case

Source abstract

We prove a quantitative polynomial Roth theorem for corners in $\mathbb F_p^2$ for arbitrary pairs of linearly independent polynomials. More precisely, given a positive integer $d$, there are constants $p_0$ and $C$ (depending only on $p$) so that for every $p>p_0$, if polynomials \(φ_1,φ_2\in \mathbb \mathbb{F}_p [y]\) are of degree $\leq d$ vanishing at $0$ and are not linearly dependent, then every $A\subset\mathbb F_p^2$ with $|A|\ge C p^{2-1/14}$ contains a nontrivial corner $$(x_1,x_2),\qquad (x_1+φ_1(y),x_2),\qquad (x_1,x_2+φ_2(y))$$ for some $y\in\mathbb F_p^\times$. This improves the estimate $p^{2-1/16}$ of Han--Lacey--Yang and removes the distinct-degree restriction from their quantitative theorem. The main obstruction is the equal-degree resonant case, where the Jacobian argument of Han--Lacey--Yang degenerates. We adjoin the frequency-independent part of the phase to form an augmented map $\widetilde F:W\to\mathbb A^3$ from the correlation threefold. We prove that this map is generically finite on every top-dimensional geometric component and has no two-dimensional fibre. Using the associated Artin--Schreier sheaf and Katz--Laumon estimates for Fourier transform of perverse sheaves, we obtain square-root cancellation outside an algebraic exceptional set of dimension at most one and uniformly bounded degree. A separate curve-sum argument gives uniform control on the exceptional set. An $\ell^2$ matrix estimate adapted to such sets completes the resonant case.