Incidence theorems for pseudoflats
Izabella Laba, Jozsef Solymosi
证明了R^n中一类曲线和R ^3中一类曲面的Pach-Sharir型关联定理，我们称之为伪平坦。特别是，我们的结果适用于广泛的一类通用的不可约的实代数集的有界度。
We prove Pach-Sharir type incidence theorems for a class of curves in R^n and surfaces in R^3, which we call pseudoflats. In particular, our results apply to a wide class of generic irreducible real algebraic sets of bounded degree.
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Absolute incidence theorems and tilings
本文提出绝对 incidence 定理的概念，并通过例子展示非绝对 incidence 定理与主定理的关系。
The paper is devoted to a proof of the de Bruijn-Erdös theorem in incidence geometry based on the Ph. Hall's marriage theorem (the theorem about the systems of distinct representatives).
We show that $m$ points and $n$ two-dimensional algebraic surfaces in $\mathbb{R}^4$ can have at most $O(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n)$ incidences, provided that the algebraic surfaces behave like pseudoflats with $k$ degrees of freedom, and that $m\leq n^{\frac{2k+2}{3k}}$. As a special case, we obtain a Szemerédi-Trotter type theorem for 2--planes in $\mathbb{R}^4$, provided $m\leq n$ and the planes intersect transversely. As a further special case, we obtain a Szemerédi-Trotter type theorem for complex lines in $\mathbb{C}^2$ with no restrictions on $m$ and $n$ (this theorem …
We improve the theorem of Beck giving a lower bound on the number of $k$-flats spanned by a set of points in real space, and improve the bound of Elekes and Tóth on the number of incidences between points and $k$-flats in real space.
Let $P = A\times A \subset \mathbb{F}_p \times \mathbb{F}_p$, $p$ a prime. Assume that $P= A\times A$ has $n$ elements, $n<p$. See $P$ as a set of points in the plane over $\mathbb{F}_p$. We show that the pairs of points in $P$ determine $\geq c n^{1 + {1/267}}$ lines, where $c$ is an absolute constant. We derive from this an incidence theorem: the number of incidences between a set of $n$ points and a set of $n$ lines in the projective plane over $\F_p$ ($n<\sqrt{p}$) is bounded by $C n^{{3/2}-{1/10678}}$, where $C$ is an absolute constant.
In the theory of line graphs of undirected graphs there exists an important theorem linking the incidence matrix of the root graph to the adjacency matrix of its line graph. For directed or mixed graphs, however, the exists no analogous result. The goal of this article is to present aligned definitions of the adjacency matrix, the incidence matrix and line graph of a mixed graph such that the mentioned theorem is valid for mixed graphs.
In this paper, we study a point-hyper plane incidence theorem in matrix rings, which generalizes all previous works in literature of this direction.
A point-variety incidence theorem over finite fields, and its applications
本文通过群代数刻画关联矩阵的奇异值与奇异向量，证明了有限域上点与簇的关联数新界，改进了有限几何中点与平坦的先前结果，并在较弱条件下扩展了钉住距离问题的已有结论。
In this note we give a shortened proof of a theorem of Rudnev, which bounds the number of incidences between points and planes over an arbitrary field. Rudnev's proof uses a map that goes via the four-dimensional Klein quadric to a three-dimensional space, where it applies a bound of Guth and Katz on intersection points of lines. We describe a simple geometric map that directly sends point-plane incidences to line-line intersections in space, allowing us to reprove Rudnev's theorem with fewer technicalities.
We examine the incidence geometry of lines in the tropical plane. We prove tropical analogs of the Sylvester-Gallai and Motzkin-Rabin theorems in classical incidence geometry. This study leads naturally to a discussion of the realizability of incidence data of tropical lines. Drawing inspiration from the von Staudt constructions and Mnëv's universality theorem, we prove that determining whether a given tropical linear incidence datum is realizable by a tropical line arrangement requires solving an arbitrary linear programming problem over the integers.
We prove almost tight bounds on incidences between points and $k$-dimensional varieties of bounded degree in $\R^d$. Our main tools are the Polynomial Ham Sandwich Theorem and induction on both the dimension and the number of points.
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 develop the methods used by Rudnev and Wheeler to prove an incidence theorem between arbitrary sets of Möbius transformations and point sets in $\mathbb F_p^2$. We also note some asymmetric incidence results, and give applications of these results to various problems in additive combinatorics and discrete geometry.
We prove an incidence result counting the $k$-rich $δ$-tubes induced by a well-spaced set of $δ$-atoms. Our result coincides with the bound that would be heuristically predicted by the Szemerédi--Trotter Theorem and holds in all dimensions $d \geq 2$.
Restrictions of incidence-preserving path maps produce an oriented hypergraphic All Minors Matrix-tree Theorems for Laplacian and adjacency matrices. The images of these maps produce a locally signed graphic, incidence generalization, of cycle covers and basic figures that correspond to incidence-k-forests. When restricted to bidirected graphs the natural partial ordering of maps results in disjoint signed boolean lattices whose minor calculations correspond to principal order ideals. As an application, (1) the determinant formula of a signed graphic Laplacian is reclaimed and shown to be dete…
We prove analogues of the Szemerédi-Trotter theorem and other incidence theorems using $δ$-tubes in place of straight lines, assuming that the $δ$-tubes are well-spaced in a strong sense.
It is shown that $n$ points and $e$ lines in the complex Euclidean plane ${\mathbb C}^2$ determine $O(n^{2/3}e^{2/3}+n+e)$ point-line incidences. This bound is the best possible, and it generalizes the celebrated theorem by Szemerédi and Trotter about point-line incidences in the real Euclidean plane ${\mathbb R}^2$.
Incidences and tilings
本文将 Pappus、Desargues 等实/复线性关联几何经典定理统一为涉及闭定向曲面四边形铺砌的单一主定理特例，为生成和推广关联定理提供通用机制。
An oriented hypergraph is an oriented incidence structure that allows for the generalization of graph theoretic concepts to integer matrices through its locally signed graphic substructure. The locally graphic behaviors are formalized in the subobject classifier of incidence hypergraphs. Moreover, the injective envelope is calculated and shown to contain the class of uniform hypergraphs -- providing a combinatorial framework for the entries of incidence matrices. A multivariable all-minors characteristic polynomial is obtained for both the determinant and permanent of the oriented hypergraphic…
We prove an incidence theorem for points and curves in the complex plane. Given a set of $m$ points in ${\mathbb R}^2$ and a set of $n$ curves with $k$ degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is $O\big(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n\big)$. We establish the slightly weaker bound $O_\varepsilon\big(m^{\frac{k}{2k-1}+\varepsilon}n^{\frac{2k-2}{2k-1}}+m+n\big)$ on the number of incidences between $m$ points and $n$ (complex) algebraic curves in ${\mathbb C}^2$ with $k$ degrees of freedom. We combine tools from algebraic geometry and differe…
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…
Multi-parameter Szemerédi-Trotter-type theorems and applications in finite fields
本文在有限域向量空间中建立了优于一般高维结果的多参数点-线关联估计，并展示了其在几何与数论组合问题中的多种应用。
We study configurations of $n$ points and $n$ lines that form $Θ(n^{4/3})$ incidences, when the point set is a Cartesian product. We prove structural properties of such configurations, such that there exist many families of parallel lines or many families of concurrent lines. We show that the line slopes have multiplicative structure or that many sets of $y$-intercepts have additive structure. We introduce the first infinite family of configurations with $Θ(n^{4/3})$ incidences. We also derive a new variant of a different structural point-line result of Elekes. Our techniques are based on the …
An $(n_3)$ configuration is an incidence structure equivalent to a linear hypergraph on $n$ vertices which is both 3-regular and 3-uniform. We investigate a variant in which one constraint, say 3-regularity, is present, and we allow exactly one line to have size four, exactly one line to have size two, and all other lines to have size three. In particular, we study planar (Euclidean or projective) representations, settling the existence question and adapting Steinitz' theorem for this setting.
In this paper we prove, for all $d \ge 2$, that for no $s<\frac{d+1}{2}$ does $I_s(μ)<\infty$ imply the canonical Falconer distance problem incidence bound, or the analogous estimate where the Euclidean norm is replaced by the norm generated by a particular convex body $B$ with a smooth boundary and everywhere non-vanishing curvature. Our construction, based on a combinatorial construction due to Pavel Valtr naturally leads us to some interesting connections between the problem under consideration, geometric incidence theorem in the discrete setting and distribution of lattice points in convex…
We prove an incidence theorem for points and planes in the projective space $\mathbb P^3$ over any field $\mathbb F$, whose characteristic $p\neq 2.$ An incidence is viewed as an intersection along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving $p$ if $p>…
Incidences, tilings, and fields
本文研究了通过铺砌证明射影几何相交定理的可能性，探讨了复数域和实数域上哪些定理可被主定理证明，引入了铺砌证明的概念，并基于拓扑空间分类了定理的层次。
Surface topology and incidence theorems over division rings
本文探讨了曲面拓扑与除环上 incidences 定理的区别，通过扩展表面图方法，证明某些定理在除环上成立，而另一些则仅在域上成立。
Incidence theorems for multivariate polynomials over finite fields
研究有限域上多元多项式的关联问题，通过谱方法证明关联三元组数量的上界，该上界在特定条件下基本精确，还推导了点与多元多项式的关联界，恢复并加强了已知结果。
Recently Guth and Katz \cite{GK2} invented, as a step in their nearly complete solution of Erdős's distinct distances problem, a new method for partitioning finite point sets in $\R^d$, based on the Stone--Tukey polynomial ham-sandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemerédi--Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at self-contained, expository treatment. We also mention some generaliza…
Elementary incidence theorems for complex numbers and quaternions
本文给出了证明以下涉及复数和四元数平面上点与线的关联的Sylvester-Gallai型定理的一些基本思想。(1)设A和B是至少各有两个复数的有限集合。则在复仿射平面中存在一条直线l，使得l与AxB相交于两点。(2)设S是复仿射平面上的有限个非共线点集。则存在一条与S相交于2、3、4或5个点的直线l。(3)设A和B是至少各有两个四元数的有限集合。则在四元数仿射平面中存在直线l，使得l与AxB相交于2、3、4或5个点。(4)设S是四元数仿射平面上的有限个非共线点集。则存在一条与S相交的直线l至少有2个至多24个点。
The Szemerédi-Trotter theorem over arbitrary field of characteristic zero
本文利用Baby Lefschetz原理将问题归约到复数情形，证明了特征零任意域上的Szemerédi-Trotter关联界，并推导出Beck定理、和积估计等推广结果。
We give an alternate proof of Schnyder's Theorem, that the incidence poset of a graph $G$ has dimension at most three if and only if $G$ is planar.
In this paper, we prove the first incidence bound for points and conics over prime fields. As applications, we prove new results on expansion of bivariate polynomial images and on certain variations of distinct distances problems. These include new lower bounds on the number of pinned algebraic distances as well as improvements of results of Koh and Sun (2014) and Shparlinski (2006) on the size of the distance set formed by two large subsets of finite dimensional vector spaces over finite fields. We also prove a variant of Beck's theorem for conics.
We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field $\mathbb{F}$, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that $m$ points and $n$ lines in $\mathbb{F}^2$, with $m^{7/8}<n<m^{8/7}$, determine at most $O(m^{11/15}n^{11/15})$ incidences (where, if $\mathbb{F}$ has positive characteristic $p$, we assume $m^{-2}n^{13}\ll p^{15}$). This improves on the previous best known bound, due to Jones. To obtain our bound, we first prove an optimal point-line incidence bound on Cartesian products, using a reduc…
We study a lower bound for the constant of the Szemerédi-Trotter theorem. In particular, we show that a recent infinite family of point-line configurations satisfies $I({\mathcal P},{\mathcal L})\ge (c+o(1)) |{\mathcal P}|^{2/3}|{\mathcal L}|^{2/3}$, with $c\approx 1.27$. Our technique is based on studying a variety of properties of Euler's totient function. We also improve the current best constant for Elekes's construction from 1 to about 1.27. From an expository perspective, this is the first full analysis of the constant of Erd\H os's construction.
Points-Polynomials Incidence Theorem with Applications to Coding Theory
本研究将有限域点线关联结果推广到点与有界次数多项式的关联，通过关联矩阵奇异值分解和群代数分析得到关联数界，并将其应用于RS码的平均半径列表译码，证明特定长度RS码可在恒定列表大小下实现该译码，填补了此前该领域的研究空白。
We present a polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean space of Guth and Katz, and its extension to hypersurfaces by Zahl and by Kaplan, Matoušek, Sharir and Safernová. We also present a bound for the number of incidences between points and hypersurfaces in the four-dimensional Euclidean space. It is an application of our partitioning theorem together with the refined bounds for the number of connected components …
In this note we generalize a recent theorem of Guth and Katz on incidences between points and lines in $3$-space from characteristic $0$ to characteristic $p$, and we explain how some of the special features of algebraic geometry in characteristic $p$ manifest themselves in problems of incidence geometry.
The perfect quantum strategies of a linear system game correspond to certain representations of its solution group. We study the solution groups of graph incidence games, which are linear system games in which the underlying linear system is the incidence system of a (non-properly) two-coloured graph. While it is undecidable to determine whether a general linear system game has a perfect quantum strategy, for graph incidence games this problem is solved by Arkhipov's theorem, which states that the graph incidence game of a connected graph has a perfect quantum strategy if and only if it either…
We study the $δ$-discretized Szemerédi-Trotter theorem and Furstenberg set problem. We prove sharp estimates for both two problems assuming tubes satisfy some spacing condition. For both two problems, we construct sharp examples that have many common features.
Inverse Theorems for Point-Sphere Incidences over Finite Fields
该研究在有限域上证明了点-球 incidences 的反向定理，揭示了近极值性与代数刚性的关系，并解决了有限域上定点距离和点积问题的结构问题。
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…
Incidence bounds in positive characteristic via valuations and distality
本文通过估值和distality证明了有限余域上的量词自由关系的distality，从而得到有限域上函数域的Szemerédi-Trotter型发生界，并推导出此类域上的Elekes-Szabó定理版本。
We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we will discuss are: (1) Counting incidences: Given a set (or several sets) of geometric objects (lines, points, etc..), what is the maximum number of incidences (or intersections) that can exist between elements in different sets? We will see several results of this type, such as the Szemeredi-Trotter theorem, over the reals and over finite fields and discuss…
We study the probability for a random line to intersect a given plane curve, defined over a finite field, in a given number of points defined over the same field. In particular, we focus on the limits of these probabilities under successive finite field extensions. Supposing absolute irreducibility for the curve, we show how a variant of Chebotarev density theorem for function fields can be used to prove the existence of these limits, and to compute them under a mildly stronger condition, known as simple tangency. Partial results have already appeared in the literature, and we propose this wor…
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 $…
We prove a theorem unifying three results from combinatorial homological and commutative algebra, characterizing the Koszul property for incidence algebras of posets and affine semigroup rings, and characterizing linear resolutions of squarefree monomial ideals. The characterization in the graded setting is via the Cohen-Macaulay property of certain posets or simplicial complexes, and in the more general nongraded setting, via the sequential Cohen-Macaulay property.
We improve the range of exponents for the restriction problem for the 3-d paraboloid over finite fields. The key new ingredient is a variant of the Bourgain-Katz-Tao finite field incidence theorem derived from sum-product estimates. In prime order fields, we give an explicit relationship between the exponent in this incidence theorem and restriction estimates for the paraboloid.
Generalized incidence theorems, homogeneous forms, and sum-product estimates in finite fields
近年来，人们利用各种组合、数论和分析方法研究了欧氏空间和有限域中的和积估计。鄂尔多斯型问题涉及的距离，面积和体积的分布也受到了广泛的关注。在本文中，我们证明了一个相对简单的功能版本的一个发病率的点和平面以前建立在\cite{HI07}和\cite{HIKR07}的结果。作为我们的方法的结果，我们得到尖锐或近尖锐的结果，由有限域上的向量空间的子集和相关的算术表达式确定的体积分布。特别是，我们的机器使我们能够证明，如果$E \subset {\Bbb F}_q^d$，$d \ge 4$，$d$维向量空间在一个有限域${\Bbb F}_q$，大小远远大于$q^{\frac{d}{2}}$，如果$E$是一个产品集，那么一组体积的$d$维平行六面体确定的$E$涵盖${\Bbb F}_q$。通过使$E$等于$A \times A \times ... \times A$可以看出，该结果是尖锐的，其中$A$是大小为$\sqrt{q}$的${\Bbb F}_q$的子场。在三维空间中，如果是$|E| \gtrsim q^{15/8}$，我们会得出相同的结果。我们证明了在三个维度上，一组卷覆盖了一个积极的比例${\Bbb F}_q$，如果$|E| \ge Cq^{3/2}$。最后，我们表明，在三维的体积集涵盖了积极的${\Bbb F}_q$的比例，如果$|E| \ge Cq^2$，没有任何进一步的假设…
Two proofs and a tight Szemerédi--Trotter theorem
本文为Lewko与Miao–Xie分别提出的两个Szemerédi–Trotter定理推广证明提供几何直观并比较，利用Hasse导数改进证明，最终得到任意代数闭域上的紧关联对数量上界。
We establish improved finite field Szemeredi-Trotter and Beck type theorems. First we show that if P and L are a set of points and lines respectively in the plane F_p^2, with |P|,|L| \leq N and N<p, then there are at most C_1 N^{3/2-1/662+o(1)} incidences between points in P and lines in L. Here C_1 is some absolute constant greater than 1. This improves on the previously best-known bound of C_1 N^{3/2-1/806+o(1)}. Second we show that if P is a set of points in \mathbb{F}_p^2 with |P|<p then either at least C_2|P|^{1-o(1)} points in P are contained in a single line, or P determines least C_2 |…
Oriented hypergraphs and generalizing the Harary-Sachs theorem to integer matrices
本文通过有向超图上的贡献者概念推广 Harary-Sachs 定理，刻画整数矩阵 Laplacian 的全子式多项式系数，并证明尾等价分组的相消性及 Laplacian 与其对偶的非零等谱性。
A planar algebraic Zarankiewicz theorem over prime fields
本文在素数域上证明了平面代数Zarankiewicz定理，给出二部图关联界，并推广至代数族，应用于富分量、差集多项式值与多项式展开。
We estimate the number of incidences in a configuration of $m$ lines and $n$ points in dimension 3. The main term is $mn^{1/3}$ if we work over the real or complex numbers but $mn^{2/5}$ over finite fields. Both of these are optimal, aside from a multiplicative constant that is at most 5.
Dimensions of Furstenberg sets and an extension of Bourgain's projection theorem
本文证明 $(s,t)$-Furstenberg 集的 Hausdorff 维数下界为 $s+t/2+\epsilon$，改进了 1999 年以来 $s<1/2$ 时的经典结果，并推广了 Bourgain 的离散化投影与乘积和定理。
Ramsey-type theorems for lines in 3-space
本文证明了三维空间中直线集合的几何 Ramsey 型定理，利用几何关联界与 Turán 型结果，给出了团或独立集的大下界，并提供了多项式时间算法。
A Szemerédi-Trotter Theorem in Arbitrary Fields
本文在任意特征域上证明Szemerédi-Trotter定理，给出点线关联数的上界，并应用于抛物面延拓估计、Bourgain提取器及sum-product估计的改进。
Discretized Radial Projections in $\mathbb{R}^d$
本文将 Orponen-Shmerkin 的 Furstenberg 型结果推广至高维，改进了 Kaufman 超平面投影定理，并提出无条件离散径向投影定理，其证明基于 $\delta$ 管与 $\delta$ 球拟积集合的新关联估计。
Bidirected graphs, integral quadratic forms and some Diophantine equations
本文研究双向图关联的非负整二次型（关联型），刻画其 Dynkin 型分类并引入 $\mathbb{C}$ 型，同时将图形框架用于研究根、丢番图方程解及 gentle 代数的 Euler 二次型。
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