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2 results

  • 13 - Incidences in Spaces Over Finite Fields

  • Adam Sheffer, Bernard M. Baruch College, City University of New York
  • Book:
    Polynomial Methods and Incidence Theory
    Published online:
    17 March 2022
    Print publication:
    24 March 2022, pp 182-203

  • Summary

    In Chapter 6 we saw several applications of polynomial methods in finite fields. In this chapter, we continue our study of finite fields, by studying point-line incidences in a finite plane. Much less is known about incidences over finite fields and many incidence problems become more difficult in this case. Unlike incidences over the reals, there is no one main technique that leads to most of the current best bounds. Instead, each bound that we derive in this chapter requires a rather different set of tools.

    For the proofs in this chapter, we introduce tools such as the projective plane, eigenvalues of a graph, and more. We also use finite field incidence bounds to study the finite field sum-product problem.