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In this paper we give a conditional improvement to the Elekes-Szab\'{o}problem over the rationals, assuming the Uniformity Conjecture. Our main resultstates that for $F\in \mathbb{Q}[x,y,z]$ belonging to a particular family ofpolynomials, 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 isalways bounded by $s \leq 5$, and so even in the worst applicable case thisgives 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 arithmeticcombinatorics. For instance, for any set $P \subset \mathbb Q^2$ and any twopoints $p_1,p_2 \in \mathbb Q^2$, 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 conditionalimprovement to a result of Sharir and Solymosi (arXiv:1308.0814).

The Elekes-Szab\'{o} Problem and the Uniformity Conjecture