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 previous 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 an $L^2$ 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.

我们证明了笛卡尔积交集大小的新 Elekes-Szabó 类型估计$A\times 乙\times C$具有代数曲面$\{F=0\}$超过实数。特别是，如果$A,乙,C$是N 个实数的集合，并且f是三变量多项式，则f具有编码加法群结构的特殊形式（例如，$F（X,y,X）=X+y-z$）， 或者$A\times 乙\times C\cap \{F=0\}$有基数$氧（{氮}^{12/7}）$。这比之前的界限有所改进$氧（{氮}^{11/6}）$。我们还证明了主要结果的非对称版本，它产生了指数为 3/2 的 Elekes-Ronyai 型扩展多项式估计。这适用于与 Erdő 不同距离问题相关的组合几何问题。

与之前解决该问题的方法一样，我们将问题改写为$L^2$估计，可以通过计算加法四元组来分析。后一个问题可以改写为涉及平面上的点和曲线的重合问题。我们证明中的新想法是，我们使用实数的顺序结构将注意力限制在较小的近似加法四元组集合上。