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.
We determine the Smith normal forms of the incidence matrices of points and projective (r-1)-dimensional subspaces of PG(n,q) and of the incidence matrices of points and r-dimensional affine subspaces of AG(n,q) for all n, r, and arbitrary prime power q.
Let $d$ and $k$ be integers with $1 \leq k \leq d-1$. Let $Λ$ be a $d$-dimensional lattice and let $K$ be a $d$-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of $k$-dimensional linear subspaces needed to cover all points in $Λ\cap K$. In particular, our results imply that the minimum number of $k$-dimensional linear subspaces needed to cover the $d$-dimensional $n \times \cdots \times n$ grid is at least $Ω(n^{d(d-k)/(d-1)-\varepsilon})$ and at most $O(n^{d(d-k)/(d-1)})$, where $\varepsilon>0$ is an arbitrarily small constant. This near…
Given positive integers $k\leq d$ and a finite field $\mathbb{F}$, a set $S\subset\mathbb{F}^{d}$ is $(k,c)$-subspace evasive if every $k$-dimensional affine subspace contains at most $c$ elements of $S$. By a simple averaging argument, the maximum size of a $(k,c)$-subspace evasive set is at most $c |\mathbb{F}|^{d-k}$. When $k$ and $d$ are fixed, and $c$ is sufficiently large, the matching lower bound $Ω(|\mathbb{F}|^{d-k})$ is proved by Dvir and Lovett. We provide an alternative proof of this result using the random algebraic method. We also prove sharp upper bounds on the size of $(k,c)$-e…
We study the permutation action of a finite symplectic group of characteristic 2 on the set of subspaces of its standard module which are either totally isotropic or else complementary to totally isotropic subspaces with respect to the alternating form. A general formula is obtained for the 2-rank of the incidence matrix for the inclusion of one-dimensional subspaces in the distinguished subspaces of a fixed dimension.
In this paper, we introduce an algebra $\mathcal{H}$ from a subspace lattice with respect to a fixed flag which contains its incidence algebra as a proper subalgebra. We then establish a relation between the algebra $\mathcal{H}$ and the quantum affine algebra $U_{q^{1/2}}(\widehat{\mathfrak{sl}}_2)$, where $q$ denotes the cardinality of the base field. It is an extension of the well-known relation between the incidence algebra of a subspace lattice and the quantum algebra $U_{q^{1/2}}(\mathfrak{sl}_2)$. We show that there exists an algebra homomorphism from $U_{q^{1/2}}(\widehat{\mathfrak{sl}…
We construct Grassmann spaces associated with the incidence geometry of regular and tangential subspaces of a symplectic copolar space, show that the underlying metric projective space can be recovered in terms of the corresponding adjacencies on so distinguished family of k-subspaces (geometrical dimension of the space being not 2k+1), and thus we prove that bijections which preserve the adjacency are determined by automorphisms of the underlying space.
In [Sharir and Solomon 2015], Sharir and Solomon showed that the number of incidences between $m$ distinct points and $n$ distinct lines in $\mathbb R^4$ is $$O^*\left(m^{2/5}n^{4/5}+ m^{1/2}n^{1/2}q^{1/4} + m^{2/3}n^{1/3}s^{1/3} + m + n\right),$$ provided that no 2-flat contains more than $s$ lines, and no hyperplane or quadric contains more than $q$ lines, where the $O^*$ hides a multiplicative factor of $2^{c\sqrt {\log m}}$ for some absolute constant $c$. In this paper we prove that, for integers $m,n,$ satisfying $n^{9/8}<m<n^{3/2}$, there exist $m$ points and $n$ lines on the quadratic h…
The Johnson graph $J(v, k)$ has as vertices the $k$-subsets of $\mathcal{V}=\{1,\ldots, v\}$, and two vertices are joined by an edge if their intersection has size $k-1$. An \emph{$X$-strongly incidence-transitive code} in $J (v, k)$ is a proper vertex subset $Γ$ such that the subgroup $X$ of graph automorphisms leaving $Γ$ invariant is transitive on the set $Γ$ of `codewords', and for each codeword $Δ$, the setwise stabiliser $X_Δ$ is transitive on $Δ\times (\mathcal{V}\setminus Δ)$. We classify the \emph{$X$-strongly incidence-transitive codes} in $J(v,k)$ for which $X$ is the symplectic gro…
Let $\mathcal{V}=\bigsqcup_{i=0}^n\mathcal{V}_i$ be the lattice of subspaces of the $n$-dimensional vector space over the finite field $\mathbb{F}_q$ and let $\mathcal{A}$ be the graded Gorenstein algebra defined over $\mathbb{Q}$ which has $\mathcal{V}$ as a $\mathbb{Q}$ basis. Let $F$ be the Macaulay dual generator for $\mathcal{A}$. We compute explicitly the Hessian determinant $|\frac{\partial ^2F}{\partial X_i \partial X_j}|$ evaluated at the point $X_1 = X_2 = \cdots = X_N=1$ and relate it to the determinant of the incidence matrix between $\mathcal{V}_1$ and $\mathcal{V}_{n-1}$. Our exp…
Let F* be the finite field of q elements and let P(n,q) be the projective space of dimension n-1 over F*. We construct a family H^{n}_{k,i} of combinatorial homology modules associated to P(n,q) over a coefficient field F field of characteristic p_{0}>0 co-prime to q. As FGL(n,q)-representations the modules are obtained from the permutation action of GL(n,q) on the subspaces of F*^n. We prove a branching rule for H^{n}_{k,i} and use this rule to determine these homology representations completely. The main results are a duality theorem and the complete characterisation of H^{n}_{k,i} in terms …
Incidence theorems for multivariate polynomials over finite fields
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学交叉科学研究院）；Frontiers Science Center for Nonlinear Expectations, Ministry of Education（教育部非线性期望前沿科学中心）；Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与交叉科学研究院）
We provide a unified approach, via deformations of incidence algebras, to several important types of representations with finiteness conditions, as well as the combinatorial algebras which produce them. We show that over finite dimensional algebras, representations with finitely many orbits, or finitely many invariant subspaces, or distributive coincide, and further coincide with thin modules in the acyclic case. Incidence algebras produce examples of such modules, and we show that algebras which are locally hereditary, and whose projective are distributive, or equivalently, which have finitel…
Let $n,n'$ be positive integers and let $V$ be an $(n+n')$-dimensional vector space over a finite field $\mathbb{F}$ equipped with a non-degenerate alternating, hermitian or quadratic form. We estimate the proportion of pairs $(U, U')$, where $U$ is a non-degenerate $n$-subspace and $U'$ is a non-degenerate $n'$-subspace of $V$, such that $U+ U'=V$ (usually such spaces $U$ and $U'$ are not perpendicular). The proportion is shown to be at least $1-c/|\mathbb{F}|$ for some constant $c\leqslant 2$ in the symplectic or unitary cases, and $c<3$ in the orthogonal case.
Let $\mathbb{F}_{q}$ be the finite field with an odd prime power $q$. In this paper, we construct a new isometric invariant of combinatorial type on $(\mathbb{F}^{n}_{q},\text{dot}_{n})$, where $\text{dot}_{n}(\mathbf{x}):=x_{1}^{2}+\cdots+x_{n}^{2}$. Additionally, using counts from our new invariant, we give a new proof of Minkowski's formula on the size of spheres over finite fields. We also show which types of quadratic subspaces can be embedded in $(\mathbb{F}_{q}^{n},\text{dot}_{n})$.
Let $L$ be a set of $n$ lines in $R^3$ that is contained, when represented as points in the four-dimensional Plücker space of lines in $R^3$, in an irreducible variety $T$ of constant degree which is \emph{non-degenerate} with respect to $L$ (see below). We show: \medskip \noindent{\bf (1)} If $T$ is two-dimensional, the number of $r$-rich points (points incident to at least $r$ lines of $L$) is $O(n^{4/3+ε}/r^2)$, for $r \ge 3$ and for any $ε>0$, and, if at most $n^{1/3}$ lines of $L$ lie on any common regulus, there are at most $O(n^{4/3+ε})$ $2$-rich points. For $r$ larger than some suffici…
On $L^2$ estimates for quadratic images of product Frostman measures
Virginia Tech（弗吉尼亚理工大学）；Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与交叉科学研究院）；Department of Mathematics, National Taiwan University（台湾大学数学系）
Let $k$ be a field, $V$ be a $k$-vector space and $X\subset V$ an algebraic irreducible subvariety. We say that a function $f:X(k) \to k$ is weakly linear if its restriction to any two-dimensional linear subspace $W$ of $V$ contained in $X$ is linear and that it is weakly quadratic if its restriction to any three-dimensional linear subspace $W$ of $V$ contained in $X$ is quadratic. We say that $X$ is admissible if any weakly linear function on $X$ is a restriction of a linear function on $V$ and any weakly quadratic function on $X$ is a restriction of a quadratic function on $V$. The main resu…
