Abstract
We use recent results about linking the number of zeros on algebraic varieties over 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 Erdős–Szemerédi conjecture in finite fields.
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Acknowledgements
The authors would like to thank Giorgis Petridis for pointing out that Theorem 2.2 is a finite field analogue of a result of Elekes and Ruzsa [8]. The authors are also grateful to Misha Rudnev for many useful comments and queries, which helped to discover a gap in the initial version.
During this work, B. K. was supported by Australian Research Council Grant DP160100932, Academy of Finland Grant 319180 and the Max Planck Institute for Mathematics, J. M. was supported by Australian Research Council Grant DP180100201, by NSERC and by the Max Planck Institute for Mathematics, and I. S. was supported by Australian Research Council Grants DP170100786 and DP180100201.
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Kerr, B., Mello, J. & Shparlinski, I.E. An effective local-global principle and additive combinatorics in finite fields. JAMA 152, 109–135 (2024). https://doi.org/10.1007/s11854-023-0291-2
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DOI: https://doi.org/10.1007/s11854-023-0291-2