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

@misc{bhakta2026systems,
  title = {{Systems of equations with elements of finitely generated groups and with linear recurrence sequences}},
  author = {Subham Bhakta and Alina Ostafe and Igor E. Shparlinski},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07227},
  eprint = {2610.07227},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07227},
}

TY  - UNPB
TI  - Systems of equations with elements of finitely generated groups and with linear recurrence sequences
AU  - Subham Bhakta
AU  - Alina Ostafe
AU  - Igor E. Shparlinski
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.07227
UR  - https://arxiv.org/abs/2610.07227
AB  - 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.
ER  -