Counting mod n in pseudofinite fields

Johnson, Will

doi:10.1007/s11856-021-2279-x

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2022

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“…If F is a pseudofinite field viewed as a structure in the language of rings, we can add a sort F n for every Galois extension of F of degree n and on each F n we add a difference operator σ n which we interpret as a generator of Gal(F n /F ). See [Joh19] for an example of this approach. Under this approach and in this restricted context, all results in this paper could be recast using the classical Lang-Weil estimates and the discussion of the counting measure found in the Appendix of [Hru02].…”

Section: Introductionmentioning

confidence: 99%

An Algebraic Hypergraph Regularity Lemma

Chevalier¹,

Levi²

2022

Preprint

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Szemerédi's regularity lemma is a powerful tool in graph theory. It states that for every large enough graph, there exists a partition of the edge set with bounded size such that most induced subgraphs are quasirandom.When the graph is a definable set φ(x, y) in a finite field F q , Tao's algebraic graph regularity lemma ([Tao12]) shows that there is a partition of the graph φ(x, y) such that all induced subgraphs are quasirandom and the error bound on quasirandomness is O(q −1/4 ).In this work we prove an algebraic hypergraph regularity lemma for definable sets in finite fields, thus answering a question of Tao. We also extend the algebraic regularity lemma to definable sets in the difference fields (F alg q , x q ) and we offer a new point of view on the geometric content of the algebraic regularity lemma.

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Section: Introductionmentioning

confidence: 99%