论文标题
使用接近度提高了Elekes-Szabó类型的估计
Improved Elekes-Szabó type estimates using proximity
论文作者
论文摘要
我们证明了关于笛卡尔产品交叉点的大小$ a \ times b \ times b \ times c $与代数表面$ \ {f = 0 \} $上的新的Elekes-Szabó类型估计。特别是，如果$ a，b，c $是$ n $实数，而$ f $是一个三元多项式，那么$ f $都具有编码添加剂组结构的特殊形式（例如$ f（x，y，x）= x + y -z $），或者$ a \ times $ a \ times b \ times b \ times b \ times c \ cap \ cap \ cap \ f = 0 \ {$ {$ {$}这是对先前绑定的$ O（n^{11/6}）$的改进。我们还证明了我们的主要结果的不对称版本，该版本可产生Elekes-Ronyai型，扩展了多项式估计，指数为$ 3/2 $。这在与ERD的不同距离问题有关的组合几何形状中有应用。 就像以前解决问题的方法一样，我们将问题重新为$ l^2 $估计，可以通过计算加法四元素来分析。后一个问题可以作为涉及平面中点和曲线的发病率问题进行重新处理。我们证明的新想法是，我们使用REAL的顺序结构将注意力限制在较小的近距离添加剂四倍体集合中。
We prove a new Elekes-Szabó type estimate on the size of the intersection of a Cartesian product $A\times B\times C$ with an algebraic surface $\{f=0\}$ over the reals. In particular, if $A,B,C$ are sets of $N$ real numbers and $f$ is a trivariate polynomial, then either $f$ has a special form that encodes additive group structure (for example $f(x,y,x) = x + y - z$), or $A \times B\times C \cap\{f=0\}$ has cardinality $O(N^{12/7})$. This is an improvement over the previously bound $O(N^{11/6})$. We also prove an asymmetric version of our main result, which yields an Elekes-Ronyai type expanding polynomial estimate with exponent $3/2$. This has applications to questions in combinatorial geometry related to the Erdős distinct distances problem. Like previous approaches to the problem, we rephrase the question as a $L^2$ estimate, which can be analyzed by counting additive quadruples. The latter problem can be recast as an incidence problem involving points and curves in the plane. The new idea in our proof is that we use the order structure of the reals to restrict attention to a smaller collection of proximate additive quadruples.
