论文标题
在$ \ mathbb f_q^2 $中计数弧线
Counting arcs in $\mathbb F_q^2$
论文作者
论文摘要
$ \ Mathbb f_q^2 $中的弧度是$ p \ subset \ mathbb f_q^2 $，因此$ p $的三个点是collinear。我们使用HyperGraph容器的方法来证明ARC的几个计数结果。令$ \ mathcal a（q）$表示$ \ mathbb f_q^2 $中的所有弧的家庭。我们的主要结果是绑定\ [| \ Mathcal a（q）| \ leq 2^{（1+O（1））Q}。 \]这与$ O（1）$符号中隐藏的因素相匹配，$（1）$ note法（考虑到所有大小$ q $的弧的所有子集）所带来的微不足道的下限。 我们还为固定（大）尺寸的弧数提供了上限。令$ k = q^t $，对于某些$ t> 2/3 $，然后$ \ mathcal a（q，k）$用$ \ mathbb f_q^2 $表示所有弧的家族，带有基数$ k $。我们证明，对于所有$γ> 0 $ \ [| \ MATHCAL A（q，k）| \ leq \ binom {（1+γ）q} {k}。 \]这个结果改善了Roche-Newton和Warren的界限。几乎匹配的下限\ [| \ Mathcal a（q，k）| \ geq \ binom {q} {k} \]通过考虑大小$ q $的弧度的所有大小$ k $的子集。
An arc in $\mathbb F_q^2$ is a set $P \subset \mathbb F_q^2$ such that no three points of $P$ are collinear. We use the method of hypergraph containers to prove several counting results for arcs. Let $\mathcal A(q)$ denote the family of all arcs in $\mathbb F_q^2$. Our main result is the bound \[ |\mathcal A(q)| \leq 2^{(1+o(1))q}. \] This matches, up to the factor hidden in the $o(1)$ notation, the trivial lower bound that comes from considering all subsets of an arc of size $q$. We also give upper bounds for the number of arcs of a fixed (large) size. Let $k=q^t$ for some $t >2/3$, and let $\mathcal A(q,k)$ denote the family of all arcs in $\mathbb F_q^2$ with cardinality $k$. We prove that, for all $γ>0$ \[ |\mathcal A(q,k)| \leq \binom{(1+γ)q}{k}. \] This result improves a bound of Roche-Newton and Warren. A nearly matching lower bound \[ |\mathcal A(q,k)| \geq \binom{q}{k} \] follows by considering all subsets of size $k$ of an arc of size $q$.
