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.
英文摘要：
We show that $m$ points and $n$ smooth algebraic surfaces of bounded degree in $\mathbb{R}^3$ satisfying suitable nondegeneracy conditions can have at most $O(m^{\frac{2k}{3k-1}}n^{\frac{3k-3}{3k-1}}+m+n)$ incidences, provided that any collection of $k$ points have at most O(1) surfaces passing through all of them, for some $k\geq 3$. In the case where the surfaces are spheres and no three spheres meet in a common circle, this implies there are $O((mn)^{3/4} + m +n)$ point-sphere incidences. This is a slight improvement over the previous bound of $O((mn)^{3/4} β(m,n)+ m +n)$ for $β(m,n)$ an (explicit) very slowly growing function. We obtain this bound by using the discrete polynomial ham sandwich theorem to cut $\mathbb{R}^3$ into open cells adapted to the set of points, and within each cell of the decomposition we apply a Turan-type theorem to obtain crude control on the number of point-surface incidences. We then perform a second polynomial ham sandwich decomposition on the irreducible components of the variety defined by the first decomposition. As an application, we obtain a new bound on the maximum number of unit distances amongst $m$ points in $\mathbb{R}^3$.
The main purpose of this paper is to prove that the point-line incidence bound due to Vinh (2011) over arbitrary finite fields can be improved in certain ranges by using tools from the VC-dimension theory. As consequences, a number of applications will be discussed in detail.
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 $d \geq 2$.
We study incidence problems involving points and curves in $R^3$. The current (and in fact only viable) approach to such problems, pioneered by Guth and Katz, requires a variety of tools from algebraic geometry, most notably (i) the polynomial partitioning technique, and (ii) the study of algebraic surfaces that are ruled by lines or, in more recent studies, by algebraic curves of some constant degree. By exploiting and refining these tools, we obtain new and improved bounds for point-curve incidence problems in $R^3$. Incidences of this kind have been considered in several previous studies, s…
We study the metric dimension and optimal split-resolving sets of the point-circle incidence graph of a Möbius plane. We prove that the metric dimension of a Möbius plane of order $q$ is around $2q$, and that an optimal split-resolving set has cardinality between approximately $5q$ and $2.5q\log q$. We also prove that a smallest blocking set of a Möbius plane of order $q$ has at most $2q(1 + \log(q + 1))$ points.
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…
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+…
We study incidences between points and algebraic curves in three dimensions, taken from a family $C$ of curves that have almost two degrees of freedom, meaning that every pair of curves intersect in $O(1)$ points, for any pair of points $p$, $q$, there are only $O(1)$ curves of $C$ that pass through both points, and a pair $p$, $q$ of points admit a curve of $C$ that passes through both of them iff $F(p,q)=0$ for some polynomial $F$. We study two specific instances, one involving unit circles in $R^3$ that pass through some fixed point (so called anchored unit circles), and the other involving…
Using degeneration and Schubert calculus, we consider the problem of computing the number of linear series of given degree $d$ and dimension $r$ on a general curve of genus $g$ satisfying prescribed incidence conditions at $n$ points. We determine these numbers completely for linear series of arbitrary dimension when $d$ is sufficiently large, and for all $d$ when either $r=1$ or $n=r+2$. Our formulas generalize and give new proofs of recent results of Tevelev and of Cela-Pandharipande-Schmitt.
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.
Orbits and incidence matrices for points, planes and lines regarding the twisted cubic in PG(3,q), q = 2, 3, 4
Kharkevich Institute for Information Transmission Problems Russian Academy of Sciences（俄罗斯科学院信息传输问题哈尔维奇研究所）；Department of Mathematics and Computer Science, University of Perugia（佩鲁贾大学数学与计算机科学系）
A celebrated unit distance conjecture due to Erd\H os says that that the unit distances cannot arise more than $C_εn^{1+ε}$ times (for any $ε>0$) among $n$ points in the Euclidean plane (see e.g. \cite{SST84} and the references contained therein). In three dimensions, the conjectured bound is $Cn^{\frac{4}{3}}$ (see e.g. \cite{KMSS12} and \cite{Z19}). In dimensions four and higher, this problem, in its general formulation, loses meaning because the Lens example shows that one can construct a set of $n$ points in dimension $4$ and higher where the unit distance arises $\approx n^2$ times (see e…
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…
