  Let $s \geq 3$ be a natural number, let $\psi(x)$ be a polynomial with real
coefficients and degree $d \geq 2$, and let $A$ be some large, non-empty,
finite subset of real numbers. We use $E_{s,2}(A)$ to denote the number of
solutions to the system of equations \[ \sum_{i=1}^{s} (\psi(x_i) -
\psi(x_{i+s}) )= \sum_{i=1}^{s} ( x_i - x_{i+s} ) = 0, \] where $x_i \in A$ for
each $1 \leq i \leq 2s$. Our main result shows that \[ E_{s,2}(A) \ll_{d,s}
|A|^{2s -3 + \eta_{s}}, \] where $\eta_3 = 1/2$, and $\eta_{s} = (1/4-
1/7246)\cdot 2^{-s + 4}$ when $s \geq 4$. The only other previously known
result of this flavour is due to Bourgain and Demeter, who showed that when
$\psi(x) = x^2$ and $s=3$, we have \[E_{3,2}(A) \ll_{\epsilon} |A|^{3 + 1/2 +
\epsilon},\] for each $\epsilon > 0$. Thus our main result improves upon the
above estimate, while also generalising it for larger values of $s$ and more
wide-ranging choices of $\psi(x)$.
  The novelty of our estimates is that they only depend on $d$, $s$ and $|A|$,
and are independent of the diameter of $A$. Thus when $A$ is a sparse set, our
results are stronger than the corresponding bounds that are provided by methods
such as decoupling and efficient congruencing. Consequently, our strategy
differs from these two lines of approach, and we employ techniques from
incidence geometry, arithmetic combinatorics and analytic number theory.
Amongst other applications, our estimates lead to stronger discrete restriction
estimates for sparse sequences.
