论文标题
有限字段上的二次子空间之间的发生率
Incidences between quadratic subspaces over finite fields
论文作者
论文摘要
令$ \ mathbb {f} _ {q} $为订单$ q $的有限字段，其中$ q $是奇数prime功率。 $（\ Mathbb {f} _ {q}^{n}，x_ {1}^{2}+x_ {2}^{2} {2}+\ cdots+cd__ {n}^2}^usom q $ q $ q $ q $ n q $ q $ q $ n q q $ n q q $（异态至$ x_ {1}^{2}+x_ {2}^{2}+\ cdots+x_ {k}^{2} $。在本文中，我们获得了$ i（\ Mathcal {k}，\ Mathcal {h}）$的发生率的界限由\ [\ left |给出i（\ Mathcal {k}，\ Mathcal {h}） - \ frac {| \ Mathcal {k} || \ Mathcal {h} |} {q^{k（n-h）}}} \ right | \ Lessim Q^{\ frac {K（2H-N-2K+4）+H（N-H-1）-2} {2}}} \ SQRT {| \ Mathcal {k} || \ Mathcal {h} | h} |}。特别是，我们改善了Phuong，Thang和Vinh（2019）获得的误差项，用于在我们其他条件下的一般收集仿射子空间。
Let $\mathbb{F}_{q}$ be a finite field of order $q$, where $q$ is an odd prime power. A quadratic subspace $(W,Q)$ of $(\mathbb{F}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})$ is called dot$_{k}$-subspace if $Q$ is isometrically isomorphic to $x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2}$. In this paper, we obtain bounds for the number of incidences $I(\mathcal{K},\mathcal{H})$ between a collection $\mathcal{K}$ of dot$_{k}$-subspaces and a collection $\mathcal{H}$ of dot$_{h}$-subspaces when $h \geq 4k-4$, which is given by \[\left | I(\mathcal{K},\mathcal{H})-\frac{|\mathcal{K}||\mathcal{H}|}{q^{k(n-h)}}\right | \lesssim q^{\frac{k(2h-n-2k+4)+h(n-h-1)-2}{2}}\sqrt{|\mathcal{K}||\mathcal{H}|}. \] In particular, we improve the error term obtained by Phuong, Thang and Vinh (2019) for general collections of affine subspaces in the presence of our additional conditions.
