论文标题
Elekes-Szabó问题和统一性猜想
The Elekes-Szabó Problem and the Uniformity Conjecture
论文作者
论文摘要
在本文中，假设统一性猜想，我们对理由的Elekes-Szabó问题有条件地改善。我们的主要结果指出，属于$ f \ in \ mathbb {q} [x，y，z] $属于一个特定的多项式家庭，任何有限的套件$ a，b，b，c \ subset \ subset \ subset \ subset \ mathbb q $ with $ | a | \ [ | z（f）\ cap（a \ times b \ times c）| \ ll n^{2- \ frac {1} {s}}。 \] 整数$ s $的值取决于多项式$ f $，但始终受$ s \ leq 5 $的限制，因此，即使在最坏的情况下，这也可以对RAZ，Sharir和de Zeeuw的界限进行定量改进（ARXIV：ARXIV：1504.05012）。 我们为离散几何和算术组合学中的问题提供了多个应用。例如，对于任何集合$ p \ subset \ mathbb q^2 $和任何两个点$ p_1，p_2 \ in \ mathbb q^2 $，我们证明，$ p_i $中的至少一个满足限制 \ [ | \ {\ | P_I -P \ | ：p \ in p \} | \ gg | p |^{3/5}， \] 其中$ \ | \ cdot \ | $表示欧几里得距离。这给出了Sharir和Solymosi的结果（Arxiv：1308.0814）的有条件改善。
In this paper we give a conditional improvement to the Elekes-Szabó problem over the rationals, assuming the Uniformity Conjecture. Our main result states that for $F\in \mathbb{Q}[x,y,z]$ belonging to a particular family of polynomials, and any finite sets $A, B, C \subset \mathbb Q$ with $|A|=|B|=|C|=n$, we have \[ |Z(F) \cap (A\times B \times C)| \ll n^{2-\frac{1}{s}}. \] The value of the integer $s$ is dependent on the polynomial $F$, but is always bounded by $s \leq 5$, and so even in the worst applicable case this gives a quantitative improvement on a bound of Raz, Sharir and de Zeeuw (arXiv:1504.05012). We give several applications to problems in discrete geometry and arithmetic combinatorics. For instance, for any set $P \subset \mathbb Q^2$ and any two points $p_1,p_2 \in \mathbb Q^2$, we prove that at least one of the $p_i$ satisfies the bound \[ | \{ \| p_i - p \| : p \in P \}| \gg |P|^{3/5}, \] where $\| \cdot \|$ denotes Euclidean distance. This gives a conditional improvement to a result of Sharir and Solymosi (arXiv:1308.0814).
