AI 中文总结
本文证明有限域上任意线性无关多项式对的定量多项式 Roth 定理，将角存在的密度阈值改进至 p^{2-1/14}，并去除不同次数限制，通过增广映射与几何方法处理等次数共振情形。
AI 中文摘要
我们证明了在 \\(\mathbb F_p^2\\) 中对于任意线性无关多项式对，一个定量多项式 Roth 定理。更精确地，给定正整数 $d$，存在常数 $p_0$ 和 $C$（仅依赖于 $d$），使得对每个 $p>p_0$，若多项式 \\(\phi_1,\phi_2\in \mathbb F_p[y]\\) 的次数不超过 $d$，在 $0$ 处取值为零且不线性相关，则每个满足 $|A|\ge C p^{2-1/14}$ 的集合 \\(A\subset\mathbb F_p^2\\) 都包含一个非平凡角 $$ (x_1,x_2),\qquad (x_1+\phi_1(y),x_2),\qquad (x_1,x_2+\phi_2(y)) $$ 其中某个 \\(y\in\mathbb F_p^\times\\)。这改进了 Han--Lacey--Yang 的估计 $p^{2-1/16}$，并去除了其定量定理中的不同次数限制。主要障碍是等次数共振情形，此时 Han--Lacey--Yang 的 Jacobian 论证退化。我们将相位的与频率无关部分附加到关联三重簇上，形成增广映射 \\(\widetilde F:W\to\mathbb A^3\\)。我们证明该映射在每个最高维几何分量上一般有限且没有二维纤维。利用相关的 Artin--Schreier 层和 Katz--Laumon 对 perverse 层 Fourier 变换的估计，我们在维数至多为一且次数一致有界的代数例外集之外获得平方根抵消。一个单独的曲线求和论证给出对例外集的一致控制。一个适应于此类集合的 \\(\ell^2\\) 矩阵估计完成了共振情形的证明。
英文摘要
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.
