有限域上的谱关联界

中文摘要 AI 辅助

设 $\Fq$ 为有限域，$F,G\in\Fq[T]$ 非常值，$J\in\Fq[T]$。我们研究参数化的双参数族 \\[ C_{a,b}=\{(x,t)\in\Fq^2:t=J(x)+aF(x)+bG(x)+\lambda ab\}, \qquad (a,b)\in\Fq^2, \\] 其中 $\lambda\in\Fq^\times$，关联按参数重数计数。若 $d=°F$，$e=°G$，且 $\charac(\Fq)>d+e$，我们证明一致谱估计 \\[ \left|I(\cP,\Omega)-\frac{|\cP||\Omega|}{q}\right| \ll_{d,e}q^{5/8}\sqrt{|\cP||\Omega|} \\] 对任意 $\cP,\Omega\subseteq\Fq^2$ 成立。证明结合两次加性傅里叶变换与Kloosterman矩阵的四阶矩估计。若 $1\le d,e\le2$ 且 $\charac(\Fq)>3$，中点差分坐标与有限Weyl量子化将误差改进至Vinh尺度 $O(q^{1/2}\sqrt{|\cP||\Omega|})$。我们进一步证明后一现象本质上是系数曲线固有的，而非所选参数化的次数所致。存在多项式Lüroth分解 \\[ F=f\circ H,\qquad G=g\circ H,\qquad \Fq(f,g)=\Fq(T), \\] 在 $H$ 的仿射变换下唯一。若 $\Gamma$ 为由 $(f,g)$ 正常参数化的平面曲线且 \\[ \kappa_H=\max_{s\in\Fq}|H^{-1}(s)|, \\] 则有限纤维拉回将中心化关联范数至多乘以 $\sqrt{\kappa_H}$。因此，若 $°\Gamma\le2$ 且 $\charac(\Fq)>3$，则 \\[ \left|I(\cP,\Omega)-\frac{|\cP||\Omega|}{q}\right| \ll \sqrt{\kappa_H}\\,q^{1/2}\sqrt{|\cP||\Omega|}, \\] 即使 $F$ 和 $G$ 具有任意大的原始次数。由此得到的 $J(x)+yF(x)+zG(x)+\lambda yz$ 的扩张估计无需对单个输入集有下界。在平衡情形下，该族的 $q^3$ 缺失值尺度已由Arala--Chow定理覆盖；此处关联定理额外在强非平衡情形下给出扩张结论。

英文摘要

Let $\Fq$ be a finite field, let $F,G\in\Fq[T]$ be nonconstant, and let $J\in\Fq[T]$. We study the parameterized two-parameter family \[ C_{a,b}=\{(x,t)\in\Fq^2:t=J(x)+aF(x)+bG(x)+λab\}, \qquad (a,b)\in\Fq^2, \] where $λ\in\Fq^\times$, with incidences counted with parameter multiplicity. If $d=°F$, $e=°G$, and $\charac(\Fq)>d+e$, we prove the uniform spectral estimate \[ \left|I(\cP,Ω)-\frac{|\cP||Ω|}{q}\right| \ll_{d,e}q^{5/8}\sqrt{|\cP||Ω|} \] for arbitrary $\cP,Ω\subseteq\Fq^2$. The proof combines two additive Fourier transforms with a fourth-moment estimate for Kloosterman matrices. If $1\le d,e\le2$ and $\charac(\Fq)>3$, midpoint--difference coordinates and finite Weyl quantization sharpen the error to the Vinh scale $O(q^{1/2}\sqrt{|\cP||Ω|})$. We further show that the latter phenomenon is intrinsic to the coefficient curve rather than to the degree of a chosen parametrization. There is a polynomial Lüroth factorization \[ F=f\circ H,\qquad G=g\circ H,\qquad \Fq(f,g)=\Fq(T), \] unique up to an affine change of $H$. If $Γ$ is the plane curve parametrized properly by $(f,g)$ and \[ κ_H=\max_{s\in\Fq}|H^{-1}(s)|, \] then finite-fibre pullback multiplies the centered incidence norm by at most $\sqrt{κ_H}$. Consequently, if $\degΓ\le2$ and $\charac(\Fq)>3$, then \[ \left|I(\cP,Ω)-\frac{|\cP||Ω|}{q}\right| \ll \sqrt{κ_H}\,q^{1/2}\sqrt{|\cP||Ω|}, \] even when $F$ and $G$ have arbitrarily large raw degree. The resulting expansion estimates for $J(x)+yF(x)+zG(x)+λyz$ hold without lower bounds on the individual input sets. In the balanced regime, the $q^3$ missing-value scale for this family is already covered by a theorem of Arala--Chow; the incidence theorems here additionally yield expansion consequences in strongly unbalanced regimes.

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