Let 𝔽q{\mathbb{F}_{q}} be the finite field of order q , where q is an odd prime power. Then a k -dimensional quadratic subspace (W,Q){(W,Q)} of (𝔽qn,x12+x22+⋯+xn2){(\mathbb{F}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})} is called dot𝐤{\operatorname{dot}_{\mathbf{k}}}-subspace if Q is isometrically isomorphic to x12+x22+⋯+xk2{x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2}}. In this paper, we obtain bounds for the number of incidences I⁢(𝒦,ℋ){I(\mathcal{K},\mathcal{H})} between a collection 𝒦{\mathcal{K}} of dotk{\operatorname{dot}_{k}}-subspaces and a collection ℋ{\mathcal{H}} of doth{\operatorname{dot}_{h}}-subspaces when h≥4⁢k-4{h\geq 4k-4}, which is given by |I⁢(𝒦,ℋ)-|𝒦|⁢|ℋ|qk⁢(n-h)|≲qk⁢(2⁢h-n-2⁢k+4)+h⁢(n-h-1)-22⁢|𝒦|⁢|ℋ|.\Bigl{\lvert}I(\mathcal{K},\mathcal{H})-\frac{\lvert\mathcal{K}\rvert\lvert% \mathcal{H}\rvert}{q^{k(n-h)}}\Bigr{\rvert}\lesssim q^{\frac{k(2h-n-2k+4)+h(n-% h-1)-2}{2}}\sqrt{\lvert\mathcal{K}\rvert\lvert\mathcal{H}\rvert}. In particular, we improve the error term in [N. D. Phuong, P. V. Thang and L. A. Vinh, Incidences between planes over finite fields, Proc. Amer. Math. Soc. 147 2019, 5, 2185–2196] obtained by Phuong, Thang and Vinh for general collections of affine subspaces in the presence of our additional conditions.

中文翻译：

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有限域上的欧几里得空间之间的入射

令 𝔽q{\mathbb{F}_{q}} 是 q 阶有限域，其中 q 是奇素数幂。然后是 (𝔽qn,x12+x22+⋯+xn2){(\mathbb{F}_{q}^{n},x_{1}^) 的 k 维二次子空间 (W,Q){(W,Q)} {2}+x_{2}^{2}+\cdots+x_{n}^{2})} 称为点𝐤{\operatorname{dot}_{\mathbf{k}}}-子空间，如果 Q 是等距的同构于 x12+x22+⋯+xk2{x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2}}。在本文中，我们获得了 dotk{\ 的集合 𝒦{\mathcal{K}} 之间的发生次数 I⁢(𝒦,ℋ){I(\mathcal{K},\mathcal{H})} 的界限operatorname{dot}_{k}}-子空间和当 h≥4⁢k-4{h\geq 4k 时 doth{\operatorname{dot}_{h}}-子空间的集合 ℋ{\mathcal{H}} -4}，由|I⁢(𝒦,ℋ)-|𝒦|⁢|ℋ|qk⁢(nh)|≲qk⁢(2⁢hn-2⁢k+4)+h⁢(nh-)给出1)-22⁢|𝒦|⁢|ℋ|.\Bigl{\lvert}I(\mathcal{K}, \mathcal{H})-\frac{\lvert\mathcal{K}\rvert\lvert% \mathcal{H}\rvert}{q^{k(nh)}}\Bigr{\rvert}\lesssim q^ {\frac{k(2h-n-2k+4)+h(n-% h-1)-2}{2}}\sqrt{\lvert\mathcal{K}\rvert\lvert\mathcal{H} \rvert}。特别是，我们改进了 [ND Phuong、PV Thang 和 LA Vinh，有限域上平面之间的入射，Proc。阿米尔。数学。社会。147 2019, 5, 2185–2196] 由 Phuong、Thang 和 Vinh 获得，用于在存在我们附加条件的情况下对仿射子空间的一般集合。