论文标题
三联和三线性指数总和的边界
Bounds of Trilinear and Trinomial Exponential Sums
论文作者
论文摘要
我们证明，对于主要残留物字段的足够小的子集$ \ MATHCAL {A} $，估计了方程$（A_1-A_2）（A_3-A_4）=（A_3-A_4）=（A_5-A_6）（A_7-A_8）（A_7-A_8）$，所有可变属于$ \ nathcal的$ \ nathcalcal} a} a}的解决方案数量。然后，我们在三联指数总和和残基总数上得出了新的界限，等于$ \ Mathcal {a} $的两个元素的乘积。我们还证明了对乘积亚组的笛卡尔产物中共线三元组的数量的完善估计，并在乘法亚组中所有变量的三连线总和来得出更强的边界。
We prove, for a sufficiently small subset $\mathcal{A}$ of a prime residue field, an estimate on the number of solutions to the equation $(a_1-a_2)(a_3-a_4) = (a_5-a_6)(a_7-a_8)$ with all variables in $\mathcal{A}$. We then derive new bounds on trilinear exponential sums and on the total number of residues equaling the product of two differences of elements of $\mathcal{A}$. We also prove a refined estimate on the number of collinear triples in a Cartesian product of multiplicative subgroups and derive stronger bounds for trilinear sums with all variables in multiplicative subgroups.
