AI 中文总结
本文针对素域上非退化二次多项式相关的和集与像集，通过中心碰撞估计等技术证明了二者的权衡关系，得到了相关指数界的结果。
AI 中文摘要
设p为奇素数，非空集合A是有限域F_p的子集且基数为N，f是F_p[x,y]中的非退化二次多项式。记S=|A+A|，M=|f(A,A)|，本文证明全范围权衡关系S⁸M⁶≳N¹⁷(1+N³/p²)⁻³，由此得max{|A+A|,|f(A,A)|}≳min{N¹⁷/¹⁴,p³/⁷N⁴/⁷}，且在N≤p²/³时指数17/14成立。证明结合了对F(u,v,w)=f(u+v,w)的中心碰撞估计、混合四能量界与求和放大，还用到两类互补关联估计：稠密碰撞区的中心谱界与稀疏区的点-平界。
英文摘要
Let $p$ be an odd prime, let $\varnothing\neq A\subseteq\mathbb F_p$ have cardinality $N$, and let $f\in\mathbb F_p[x,y]$ be a non-degenerate quadratic polynomial. Writing $S=|A+A|$ and $M=|f(A,A)|$, we prove the full-range trade-off $S^8M^6\gtrsim N^{17}(1+N^3/p^2)^{-3}$. Consequently, $\max\{|A+A|,|f(A,A)|\}\gtrsim \min\{N^{17/14},p^{3/7}N^{4/7}\}$, and in particular the exponent $17/14$ holds throughout $N\le p^{2/3}$. The proof combines a centered collision estimate for $F(u,v,w)=f(u+v,w)$, a mixed fourth-energy bound, and a popular-sum amplification. Two complementary incidence estimates enter the argument: a centered spectral bound in the dense collision regime and a point--plane bound in the sparse regime.
Comments10 pages. No figures
