An effective local-global principle for algebraic varieties and the sum product problem in finite fields

Bryce Kerr; Jorge Mello; Igor Shparlinski

An effective local-global principle for algebraic varieties and the sum product problem in finite fields

Number Theory 2022-07-25 v3

Authors: Bryce Kerr , Jorge Mello , Igor Shparlinski

Abstract

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.

Keywords

local-global principle algebraic varieties sum product problem finite fields linear forms congruences

Cite

@article{arxiv.2005.02923,
  title  = {An effective local-global principle for algebraic varieties and the sum product problem in finite fields},
  author = {Bryce Kerr and Jorge Mello and Igor Shparlinski},
  journal= {arXiv preprint arXiv:2005.02923},
  year   = {2022}
}

Comments

Corrected an error in the previous version and added a stronger result which holds for almost all primes

An effective local-global principle for algebraic varieties and the sum product problem in finite fields

Abstract

Keywords

Cite

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