Systems of equations with elements of finitely generated groups and with linear recurrence sequences

Abstract

We obtain new and essentially optimal bounds on the number of solutions of diagonal systems of $k$ polynomial equations in $s$ variables from an arbitrary finite subset of a finitely generated group in a field of characteristic zero. Our bounds generalise and sometimes improve previous results of J. Bourgain, M. Z. Garaev, S. V. Konyagin, and I. E. Shparlinski (2014), for $k=1$, and of A. Manning, A. Ostafe and I. E. Shparlinski (2026), for $k=2$. In turn, this improvement leads to sharpening a result of A. Manning, A. Ostafe and I. E. Shparlinski (2026) concerning arithmetic statistics of certain matrices. We also obtain similar results for $s$-tuples of linear recurrence sequences, in the case of linear equations and quadratic equations.