多项式大小的三角网格上的多种序类型
Many Order Types on Integer Grids of Polynomial Size
摘要：
如果对于每个$i,j,k$，三元组$(p_i,p_j,p_k)$和$(q_i,q_j,q_k)$具有相同的方向，则两个标记点配置$\{p_1,\ldots,p_n\}$和$\{q_1,\ldots,q_n\}$具有相同的顺序类型。在20世纪80年代，Goodman，Pollack和Sturmfels证明了（i）$n$点上的序类型的数量是$4^{n+o(n)}$阶，（ii）所有的序类型都可以用双指数整数坐标来实现，以及（iii）某些序类型确实需要双指数整数坐标。在2018年，Caraballo，Díaz-Bá{}ez，Fabila-Monroy，Hidalgo-Toscano，Lea{}os，Montejano证明了至少$n^{3n+o(n)}$阶类型可以在多项式大小的整数网格上实现。在这篇文章中，我们改进了他们的结果，证明了至少$n^{4n+o(n)}$阶类型可以在多项式大小的整数网格上实现，这基本上是最好的。
英文摘要：
Two labeled point configurations $\{p_1,\ldots,p_n\}$ and $\{q_1,\ldots,q_n\}$ are of the same order type if, for every $i,j,k$, the triples $(p_i,p_j,p_k)$ and $(q_i,q_j,q_k)$ have the same orientation. In the 1980's, Goodman, Pollack and Sturmfels showed that (i) the number of order types on $n$ points is of order $4^{n+o(n)}$, (ii) all order types can be realized with double-exponential integer coordinates, and that (iii) certain order types indeed require double-exponential integer coordinates. In 2018, Caraballo, Díaz-Bá{ñ}ez, Fabila-Monroy, Hidalgo-Toscano, Lea{ñ}os, Montejano showed that at least $n^{3n+o(n)}$ order types can be realized on an integer grid of polynomial size. In this article, we improve their result by showing that at least $n^{4n+o(n)}$ order types can be realized on an integer grid of polynomial size, which is essentially best possible.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
