Let $\mathbb{F}_q$ be a finite field of $q$ elements where $q$ is a large odd prime power and $Q =a_1 x_1^{c_1}+...+a_dx_d^{c_d}\in \mathbb{F}_q[x_1,...,x_d]$, where $2\le c_i\le N$, $\gcd(c_i,q)=1$, and $a_i\in \mathbb{F}_q$ for all $1\le i\le d$. A $Q$-sphere is a set of the form $\lbrace x\in \mathbb{F}_q^d | Q(x-b)=r\rbrace$, where $b\in \mathbb{F}_q^d, r\in \mathbb{F}_q$. We prove bounds on the number of incidences between a point set $\mathcal{P}$ and a $Q$-sphere set $\mathcal{S}$, denoted by $I(\mathcal{P},\mathcal{S})$, as the following.
$$| I(\mathcal{P},\mathcal{S})-\frac{|\mathcal{P}||\mathcal{S}|}{q}|\le q^{d/2}\sqrt{|\mathcal{P}||\mathcal{S}|}.$$
We prove this estimate by studying the spectra of directed graphs. We also give a version of this estimate over finite rings $\mathbb{Z}_q$ where $q$ is an odd integer. As a consequence of the above bounds, we give an estimate for the pinned distance problem. In Sections $4$ and $5$, we prove a bound on the number of incidences between a random point set and a random $Q$-sphere set in $\mathbb{F}_q^d$. We also study the finite field analogues of some combinatorial geometry problems, namely, the number of generalized isosceles triangles, and the existence of a large subset without repeated generalized distances.
In this paper, we prove a new point-sphere incidence bound in vector spaces over finite fields. More precisely, let $P$ be a set of points and $S$ be a set of spheres in $\mathbb{F}_q^d$. Suppose that $|P|, |S|\le N$, we prove that the number of incidences between $P$ and $S$ satisfies \[I(P, S)\le N^2q^{-1}+q^{\frac{d-1}{2}}N,\] under some conditions on $d, q$, and radii. This improves the known upper bound $N^2q^{-1}+q^{\frac{d}{2}}N$ in the literature. As an application, we show that for $A\subset \mathbb{F}_q$ with $q^{1/2}\ll |A|\ll q^{\frac{d^2+1}{2d^2}}$, one has \[\max \left\lbrace |A+…
In this paper we show that the number of distinct distances determined by a set of $n$ points on a constant-degree two-dimensional algebraic variety $V$ (i.e., a surface) in $\mathbb R^3$ is at least $Ω\left(n^{7/9}/{\rm polylog} \,n\right)$. This bound is significantly larger than the conjectured bound $Ω(n^{2/3})$ for general point sets in $\mathbb R^3$. We also show that the number of unit distances determined by $n$ points on a surface $V$, as above, is $O(n^{4/3})$, a bound that matches the best known planar bound, and is worst-case tight in 3-space. This is in sharp contrast with the bes…
We use elementary methods to prove an incidence theorem for points and spheres in $\mathbb{F}_q^n$. As an application, we show that any point set of $P\subset \mathbb{F}_q^2$ with $|P|\geq 5q$ determines a positive proportion of all circles. The latter result is an analogue of Beck's Theorem for circles which is optimal up to multiplicative constants.
We present a technique for deriving lower bounds for incidences with hypersurfaces in ${\mathbb R}^d$ with $d\ge 4$. These bounds apply to a large variety of hypersurfaces, such as hyperplanes, hyperspheres, paraboloids, and hypersurfaces of any degree. Beyond being the first non-trivial lower bounds for various incidence problems, our bounds show that some of the known upper bounds for incidence problems in ${\mathbb R}^d$ are tight up to an extra $\varepsilon$ in the exponent. Specifically, for every $m$, $d\ge 4$, and $\varepsilon>0$ there exist $m$ points and $n$ hypersurfaces in ${\mathbb…
An improved bound on the number of point-surface incidences in three dimensions
我们证明了满足适当的非退化条件的$\mathbb{R}^3$中有界度的$m$点和$n$光滑代数曲面最多有$O(m^{\frac{2k}{3k-1}}n^{\frac{3k-3}{3k-1}}+m+n)$关联，只要对某些$k\geq 3$，任何$k$点的集合最多有O（1）个曲面通过它们.在曲面为球面且没有三个球面相交于同一圆的情况下，这意味着存在$O((mn)^{3/4} + m +n)$点-球面入射。对于$β(m,n)$（显式）增长非常缓慢的函数，这比之前的$O((mn)^{3/4} β(m,n)+ m +n)$边界略有改进。我们得到这个界限使用离散多项式火腿三明治定理切割$\mathbb{R}^3$到开放的细胞适应的点集，并在每个细胞的分解，我们应用图兰型定理，以获得粗糙的控制点表面的发病率。然后，我们执行第二个多项式火腿三明治分解的不可约组件的品种所定义的第一次分解。作为应用，我们得到了$\mathbb{R}^3$中$m$点间最大单位距离个数的一个新的界.
In this paper we introduce a unified approach to deal with incidence problems between points and varieties over finite fields. More precisely, we prove that the number of incidences $I(\mathcal{P}, \mathcal{V})$ between a set $\mathcal{P}$ of points and a set $\mathcal{V}$ of varieties of a certain form satisfies $$\left\vert I(\mathcal{P},\mathcal{V})-\frac{|\mathcal{P}||\mathcal{V}|}{q^k}\right\vert\le q^{dk/2}\sqrt{|\mathcal{P}||\mathcal{V}|}.$$ This result is a generalization of the results of Vinh (2011), Bennett et al. (2014), and Cilleruelo et al. (2015). As applications of our incidenc…
There is a convolution product on 3-variable partial flag functions of a locally finite poset that produces a generalized Möbius function. Under the product this generalized Möbius function is a one sided inverse of the zeta function and satisfies many generalizations of classical results. In particular we prove analogues of Phillip Hall's Theorem on the Möbius function as an alternating sum of chain counts, Weisner's theorem, and Rota's Crosscut Theorem. A key ingredient to these results is that this function is an overlapping product of classical Möbius functions. Using this generalized Möbi…
Non-degeneracy was first defined for hyperplanes by Elekes-Tóth, and later extended to spheres by Apfelbaum-Sharir: given a set $P$ of $m$ points in $\mathbb{R}^d$ and some $β\in(0,1)$, a $(d-1)$-dimensional sphere (or a $(d-1)$-sphere) $S$ in $\mathbb{R}^d$ is called $β$-nondegenerate with respect to $P$ if $S$ does not contain a proper subsphere $S'$ such that $|S'\cap P|\geq β|S\cap P|$. Apfelbaum-Sharir found an upper bound for the number of incidences between points and nondegenerate spheres in three dimensions, which was recently used by Zahl to obtain the best known bound for the unit d…
A generalization of incidence relations in abstract polytope has been explored, and parameterized surfaces are used as primers. The abstract orientable incidence structure is defined as an algebraic model of incidence relations, in which some algebraic properties in abtract polytope theory are generalized. The geometric interpretation of abstract orientable incidence structure are also discussed. The orientable incidence structure in a semi-regular normal CW complex are briefly investigated.
In this paper we study the boundedness of extension operators associated with spheres in vector spaces over finite fields.In even dimensions, we estimate the number of incidences between spheres and points in the translated set from a subset of spheres. As a result, we improve the Tomas-Stein exponents, our previous results. The analytic approach and the explicit formula for Fourier transform of the characteristic function on spheres play an important role to get good bounds for exponential sums.
We study valuated matroids, their tropical incidence relations, flag matroids and total positivity. This leads to a characterization of permutahedral subdivisions, namely subdivisions of regular permutahedra into generalized permutahedra. Further, we get a characterization of those subdivisions arising from positive valuated flag matroids.
A tetravalent $2$-arc-transitive graph of order $728$ is either the known $7$-arc-transitive incidence graph of the classical generalized hexagon $GH(3,3)$ or a normal cover of a $2$-transitive graph of order $182$ denoted $A[182,1]$ or $A[182,2]$ in the $2009$ list of Potočnik.
The article studies power complexes and generalized power complexes, and investigates the algebraic structure of their automorphism groups. The combinatorial incidence structures involved are cube-like, in the sense that they have many structural properties in common with higher-dimensional cubes and cubical tessellations on manifolds. Power complexes have repeatedly appeared in applications.
The point-plane incidence theorem states that the number of incidences between $n$ points and $m\geq n$ planes in the projective three-space over a field $F$, is $$O\left(m\sqrt{n}+ m k\right),$$ where $k$ is the maximum number of collinear points, with the extra condition $n< p^2$ if $F$ has characteristic $p>0$. This theorem also underlies a state-of-the-art Szemerédi-Trotter type bound for point-line incidences in $F^2$, due to Stevens and de Zeeuw. This review focuses on some recent, as well as new, applications of these bounds that lead to progress in several open geometric questions in $…
This paper is concerned with the affine-invariant ternary codes which are defined by Hermitian functions. We compute the incidence matrices of 2-designs that are supported by the minimum weight codewords of these ternary codes. The linear codes generated by the rows of these incidence matrix are subcodes of the extended codes of the 4-th order generalized Reed-Muller codes and they also hold 2-designs. Finally, we give the dimensions and lower bound of the minimum weights of these linear codes.
A Bier sphere $Bier(K) = K\ast_ΔK^\circ$, defined as the deleted join of a simplicial complex and its Alexander dual $K^\circ$, is a purely combinatorial object (abstract simplicial complex). Here we study a hidden geometry of Bier spheres by describing their natural geometric realizations, compute their volume, describe an effective criterion for their polytopality, and associate to $K$ a natural fan $Fan(K)$, related to the Braid fan. Along the way we establish a connection of Bier spheres of maximal volume with recent generalizations of the classical Van Kampen-Flores theorem and clarify th…
We generalize the Szemerédi-Trotter incidence theorem, to bound the number of complete \emph{flags} in higher dimensions. Specifically, for each $i=0,1,\ldots,d-1$, we are given a finite set $S_i$ of $i$-flats in $\R^d$ or in $\C^d$, and a (complete) flag is a tuple $(f_0,f_1,\ldots,f_{d-1})$, where $f_i\in S_i$ for each $i$ and $f_i\subset f_{i+1}$ for each $i=0,1,\ldots,d-2$. Our main result is an upper bound on the number of flags which is tight in the worst case. We also study several other kinds of incidence problems, including (i) incidences between points and lines in $\R^3$ such that a…
A generalized quadrangle is a point-line incidence geometry $\mathcal{Q}$ such that: (i) any two points lie on at most one line, and (ii) given a line $\ell$ and a point $P$ not incident with $\ell$, there is a unique point of $\ell$ collinear with $P$. The finite Moufang generalized quadrangles were classified by Fong and Seitz (1973), and we study a larger class of generalized quadrangles: the \emph{antiflag-transitive} quadrangles. An antiflag of a generalized quadrangle is a non-incident point-line pair $(P, \ell)$, and we say that the generalized quadrangle $\mathcal{Q}$ is antiflag-trans…
This paper is a survey of recent advances as well as open problems in the study of face numbers of centrally symmetric simplicial polytopes and spheres. The topics discussed range from neighborliness of centrally symmetric polytopes and the upper bound theorem for centrally symmetric simplicial spheres to the generalized lower bound theorem for centrally symmetric simplicial polytopes and the lower bound conjecture for centrally symmetric simplicial spheres and manifolds.
Let $G$ be a connected graph and let $\mathbb{X}$ be the set of projective points defined by the column vectors of the incidence matrix of $G$ over a field $K$ of any characteristic. We determine the generalized Hamming weights of the Reed--Muller-type code over the set $\mathbb{X}$ in terms of graph theoretic invariants. As an application to coding theory we show that if $G$ is non-bipartite and $K$ is a finite field of ${\rm char}(K)\neq 2$, then the $r$-th generalized Hamming weight of the linear code generated by the rows of the incidence matrix of $G$ is the $r$-th weak edge biparticity o…
We discuss a unified approach to a class of geometric combinatorics incidence problems in $2D$, of the Erdös distance type. The goal is obtaining the second moment estimate, that is given a finite point set $S$ and a function $f$ on $S\times S$, an upper bound on the number of solutions of $$ f(p,p') = f(q,q')\neq 0,\qquad (p,p',q,q')\in S\times S\times S\times S. \qquad(*) $$ E.g., $f$ is the Euclidean distance in the plane, sphere, or a sheet of the two-sheeted hyperboloid. Our tool is the Guth-Katz incidence theorem for lines in $\mathbb{RP}^3$, but we focus on how the original $2D$ problem…
We study generalized eigenvalue problems for meet and join matrices with respect to incidence functions on semilattices. We provide new bounds for generalized eigenvalues of meet matrices with respect to join matrices under very general assumptions. The applied methodology is flexible, and it is shown in the case of GCD and LCM matrices that even sharper bounds can be obtained by applying the known properties of the divisor lattice. These results can also be easily modified for the dual problem of eigenvalues of join matrices with respect to meet matrices, which we briefly consider as well. We…
We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov '03] established that if the clause-variable incidence graph of a CNF formula F is a good enough expander, then proving that F is unsatisfiable requires high PC/PCR degree. We further develop the techniques in [AR03] to show that if one can "cluster" clauses and variables in a way that "respects the structure" of the formula in a certain sense, then it is sufficient that the inci…
