  We use recent results about linking the number of zeros on algebraic
varieties over $\mathbb{C}$, defined by polynomials with integer coefficients,
and on their reductions modulo sufficiently large primes to study congruences
with products and reciprocals of linear forms. This allows us to make some
progress towards a question of B. Murphy, G. Petridis, O. Roche-Newton, M.
Rudnev and I. D. Shkredov (2019) on an extreme case of the
Erd\H{o}s-Szemer\'{e}di conjecture in finite fields.
