An Algebraic Hypergraph Regularity Lemma

Szemer\'edi'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 $\phi(x, y)$ in a finite field $F_q$, Tao's algebraic graph regularity lemma shows that there is a partition of the graph $\phi(x, y)$ 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 $(F_q^{alg}, x^q)$ and we offer a new point of view on the geometric content of the algebraic regularity lemma.