论文标题
代数超晶规则性引理
An Algebraic Hypergraph Regularity Lemma
论文作者
论文摘要
Szemerédi的规律性引理是图理论中的强大工具。它指出，对于每个足够大的图形，都存在具有有界尺寸的边缘设置的分区，以使大多数引起的子图是quasirandom。当图是有限字段中的$ ϕ（x，y）$ f_q $的可定义集$ f_q $时，陶的代数图规则性引理表明，图$ ϕ（x，y）$的分区，使所有诱导的子图是quasirandom is quasirandom is quasirandom is quasirandomness in quasirandomness is quasirandomness is of quasirandomness is $ o o（q^^{q^{-1/44}）。在这项工作中，我们证明了在有限领域可定义的集合的代数超法规律性引理，从而回答了陶的问题。我们还将代数规则性引理扩展到差异字段$（f_q^{alg}，x^q）$中的可定义集，我们为代数规律性引理的几何内容提供了新的观点。
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 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 $(F_q^{alg}, x^q)$ and we offer a new point of view on the geometric content of the algebraic regularity lemma.
