  In this paper we give a conditional improvement to the Elekes-Szab\'{o}
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).
