arXiv subjects

Frank de Zeeuw

Publications and source records attributed to Frank de Zeeuw.

At least 19 recordsLinked to original sources

arXiv2022

A Sylvester-Gallai theorem for cubic curves

We prove a variant of the Sylvester-Gallai theorem for cubics (algebraic curves of degree three): If a finite set of sufficiently many points in $\mathbb{R}^2$ is not contained in a cubic, then there is a cubic that contains exactly nine of the points. This resolves the first unknown case of a conjecture of Wiseman and Wilson from 1988, who proved a variant of Sylvester-Gallai for conics and conjectured that similar statements hold for curves of any degree.

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arXiv2018

Constructions for the Elekes-Szabó and Elekes-Rónyai problems

We give a construction of a non-degenerate polynomial $F\in \mathbb R[x,y,z]$ and a set $A$ of cardinality $n$ such that $\left|Z(F)\cap (A \times A \times A) \right| \gg n^{\frac{3}{2}}$, thus providing a new lower bound construction for the Elekes--Szabó problem. We also give a related construction for the Elekes--Rónyai problem restricted to a subgraph. This consists of a polynomial $f\in \mathbb R[x,y]$ that is not additive or multiplicative, a set $A$ of size $n$, and a subset $P\subset A\times A$ of size $|P|\gg n^{3/2}$ on which $f$ takes only $n$ distinct values.

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arXiv2018

On sets defining few ordinary circles

An ordinary circle of a set $P$ of $n$ points in the plane is defined as a circle that contains exactly three points of $P$. We show that if $P$ is not contained in a line or a circle, then $P$ spans at least $\frac{1}{4}n^2 - O(n)$ ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large $n$ and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that $P$ spans at most $\frac{1}{24}n^3 - O(n^2)$ circles passing through exactly four points of $P$. Here we determine the exact maximum and the extremal configurations for all sufficiently large $n$. These results are based on the following structure theorem. If $n$ is sufficiently large depending on $K$, and $P$ is a set of $n$ points spanning at most $Kn^2$ ordinary circles, then all but $O(K)$ points of $P$ lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.

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arXiv2018

Ordinary lines in space

We prove that if a finite point set in real space does not have too many points on a plane, then it spans a quadratic number of ordinary lines. This answers the real case of a question of Basit, Dvir, Saraf, and Wolf. It shows that there is a significant difference in terms of ordinary lines between planar point sets, which may span a linear number of ordinary lines, and truly three-dimensional point sets. Our proof uses a projection argument of Kelly combined with a theorem of Beck on the number of spanned lines of a planar point set.

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arXiv2018

Spanned lines and Langer's inequality

We collect some results in combinatorial geometry that follow from an inequality of Langer in algebraic geometry. Langer's inequality gives a lower bound on the number of incidences between a point set and its spanned lines, and was recently used by Han to improve the constant in the weak Dirac conjecture. Here we observe that this inequality also leads to improved constants in Beck's theorem, which states that a finite point set in the real or complex plane has many points on a line or spans many lines. Most of the proofs that we use are not original, and the goal of this note is mainly to carefully record the quantitative results in one place. We also include some discussion of possible further improvements to these statements.

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arXiv2017

Schwartz-Zippel bounds for two-dimensional products

We prove bounds on intersections of algebraic varieties in $\mathbb{C}^4$ with Cartesian products of finite sets from $\mathbb{C}^2$, and we point out connections with several classic theorems from combinatorial geometry. Consider an algebraic variety $X$ in $\mathbb{C}^4$ of degree $d$, such that the polynomials defining $X$ are not all of the form $F(x,y,s,t) = G(x,y)H(x,y,s,t) + K(s,t)L(x,y,s,t)$. Let $P$ and $Q$ be finite subsets of $\mathbb{C}^2$ of size $n$. If $X$ has dimension one or two, then we prove $|X\cap (P\times Q)| = O_d(n)$, while if $X$ has dimension three, then $|X\cap (P\times Q)| =O_{d,\varepsilon}(n^{4/3+\varepsilon})$ for any $\varepsilon>0$. Both bounds are best possible in this generality (except for the $\varepsilon$). These bounds can be viewed as different generalizations of the Schwartz-Zippel lemma, where we replace a product of "one-dimensional" finite subsets of $\mathbb{C}$ by a product of "two-dimensional" finite subsets of $\mathbb{C}^2$. The bound for three-dimensional varieties generalizes the Szemerédi-Trotter theorem. A key ingredient in our proofs is a two-dimensional version of a special case of Alon's combinatorial Nullstellensatz. As corollaries of our two bounds, we obtain bounds on the number of repeated and distinct values of polynomials and polynomial maps of pairs of points in $\mathbb{C}^2$, with a characterization of those maps for which no good bounds hold. These results generalize known bounds on repeated and distinct Euclidean distances.

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arXiv2017

An Improved Point-Line Incidence Bound Over Arbitrary Fields

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 reduction to a point-plane incidence bound of Rudnev. We then cover most of the point set with Cartesian products, and we bound the incidences on each product separately, using the bound just mentioned. We give several applications, to sum-product-type problems, an expander problem of Bourgain, the distinct distance problem and Beck's theorem.

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arXiv2017

Three-variable expanding polynomials and higher-dimensional distinct distances

We determine which quadratic polynomials in three variables are expanders over an arbitrary field $\mathbb{F}$. More precisely, we prove that for a quadratic polynomial $f\in \mathbb{F}[x,y,z]$, which is not of the form $g(h(x)+k(y)+l(z))$, we have $|f(A\times B\times C)|\gg N^{3/2}$ for any sets $A,B,C\subset \mathbb{F}$ with $|A|=|B|=|C|=N$, with $N$ not too large compared to the characteristic of $\mathbb{F}$. We give several applications. We use this result for $f=(x-y)^2+z$ to obtain new lower bounds on $|A+A^2|$ and $\max\{|A+A|,|A^2+A^2|\}$, and to prove that a Cartesian product $A\times\cdots \times A\subset \mathbb{F}^d$ determines almost $|A|^2$ distinct distances if $|A|$ is not too large.

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arXiv2016

A short proof of Rudnev's point-plane incidence bound

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.

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arXiv2016

The Elekes-Szabó Theorem in four dimensions

Let $F\in\mathbb{C}[x,y,s,t]$ be an irreducible constant-degree polynomial, and let $A,B,C,D\subset\mathbb{C}$ be finite sets of size $n$. We show that $F$ vanishes on at most $O(n^{8/3})$ points of the Cartesian product $A\times B\times C\times D$, unless $F$ has a special group-related form. A similar statement holds for $A,B,C,D$ of unequal sizes. This is a four-dimensional extension of our recent improved analysis of the original Elekes-Szabó theorem in three dimensions. We give three applications: an expansion bound for three-variable real polynomials that do not have a special form, a bound on the number of coplanar quadruples on a space curve that is neither planar nor quartic, and a bound on the number of four-point circles on a plane curve that has degree at least five.

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arXiv2016

On the number of ordinary conics

We prove a lower bound on the number of ordinary conics determined by a finite point set in $\mathbb{R}^2$. An ordinary conic for a subset $S$ of $\mathbb{R}^2$ is a conic that is determined by five points of $S$, and contains no other points of $S$. Wiseman and Wilson proved the Sylvester-Gallai-type statement that if a finite point set is not contained in a conic, then it determines at least one ordinary conic. We give a simpler proof of their result and then combine it with a result of Green and Tao to prove our main result: If $S$ is not contained in a conic and has at most $c|S|$ points on a line, then $S$ determines $Ω_c(|S|^4)$ ordinary conics. We also give a construction, based on the group structure of elliptic curves, that shows that the exponent in our bound is best possible.

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arXiv2016

On the number of ordinary circles

We prove that any $n$ points in $\mathbb{R}^2$, not all on a line or circle, determine at least $\frac{1}{4}n^2-O(n)$ ordinary circles (circles containing exactly three of the $n$ points). The main term of this bound is best possible for even $n$. Our proof relies on a recent result of Green and Tao on ordinary lines.

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arXiv2016

A survey of Elekes-Rónyai-type problems

We give an overview of recent progress around a problem introduced by Elekes and Rónyai. The prototype problem is to show that a polynomial $f\in \mathbb{R}[x,y]$ has a large image on a Cartesian product $A\times B\subset \mathbb{R}^2$, unless $f$ has a group-related special form. We discuss a number of variants and generalizations. This includes the Elekes-Szabó problem, which generalizes the Elekes-Rónyai problem to a question about an upper bound on the intersection of an algebraic surface with a Cartesian product, and curve variants, where we ask the same questions for Cartesian products of finite subsets of algebraic curves. These problems lie at the crossroads of combinatorics, algebra, and geometry: They ask combinatorial questions about algebraic objects, whose answers turn out to have applications to geometric questions involving basic objects like distances, lines, and circles, as well as to sum-product-type questions from additive combinatorics. As part of a recent surge of algebraic techniques in combinatorial geometry, a number of quantitative and qualitative steps have been made within this framework. Nevertheless, many tantalizing open questions remain.

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arXiv2015

Distinct distances between points and lines

We show that for $m$ points and $n$ lines in the real plane, the number of distinct distances between the points and the lines is $Ω(m^{1/5}n^{3/5})$, as long as $m^{1/2}\le n\le m^2$. We also prove that for any $m$ points in the plane, not all on a line, the number of distances between these points and the lines that they span is $Ω(m^{4/3})$. The problem of bounding the number of distinct point-line distances can be reduced to the problem of bounding the number of tangent pairs among a finite set of lines and a finite set of circles in the plane, and we believe that this latter question is of independent interest. In the same vein, we show that $n$ circles in the plane determine at most $O(n^{3/2})$ points where two or more circles are tangent, improving the previously best known bound of $O(n^{3/2}\log n)$. Finally, we study three-dimensional versions of the distinct point-line distances problem, namely, distinct point-line distances and distinct point-plane distances. The problems studied in this paper are all new, and the bounds that we derive for them, albeit most likely not tight, are non-trivial to prove. We hope that our work will motivate further studies of these and related problems.

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arXiv2015

Incidence bounds for complex algebraic curves on Cartesian products

We prove bounds on the number of incidences between a set of algebraic curves in $\mathbb{C}^2$ and a Cartesian product $A\times B$ with finite sets $A,B\subset \mathbb{C}$. Similar bounds are known under various conditions, but we show that the Cartesian product assumption leads to a simpler proof. This assumption holds in a number of interesting applications, and with our bound these applications can be extended from $\mathbb{R}$ to $\mathbb{C}$. The proof is a new application of the polynomial partitioning technique introduced by Guth and Katz.

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arXiv2015

Distinct distances on algebraic curves in the plane

Let $P$ be a set of $n$ points in the real plane contained in an algebraic curve $C$ of degree $d$. We prove that the number of distinct distances determined by $P$ is at least $c_d n^{4/3}$, unless $C$ contains a line or a circle. We also prove the lower bound $c_d' \min(m^{2/3}n^{2/3}, m^2, n^2)$ for the number of distinct distances between $m$ points on one irreducible plane algebraic curve and $n$ points on another, unless the two curves are parallel lines, orthogonal lines, or concentric circles. This generalizes a result on distances between lines of Sharir, Sheffer, and Solymosi in arXiv:1302.3081.

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arXiv2015

Polynomials vanishing on Cartesian products: The Elekes-Szabó Theorem revisited

Let $F\in\mathbb{C}[x,y,z]$ be a constant-degree polynomial,and let $A,B,C\subset\mathbb C$ be finite sets of size $n$. We show that $F$ vanishes on at most $O(n^{11/6})$ points of the Cartesian product $A\times B\times C$, unless $F$ has a special group-related form. This improves a theorem of Elekes and Szabó [Combinatorica, 2012], and generalizes a result of Raz, Sharir, and Solymosi [Amer. J. Math., to appear]. The same statement holds over $\mathbb{R}$, and a similar statement holds when $A, B, C$ have different sizes (with a more involved bound replacing $O(n^{11/6})$). This result provides a unified tool for improving bounds in various Erd\H os-type problems in combinatorial geometry, and we discuss several applications of this kind.

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arXiv2015

Distinct values of bilinear forms on algebraic curves

Let $B$ be a bilinear form on pairs of points in the complex plane, of the form $B(p,q) = p^TMq$, for an invertible $2\times2$ complex matrix $M$. We prove that any finite set $S$ contained in an irreducible algebraic curve $C$ of degree $d$ in $\mathbb{C}^2$ determines at least $c_d|S|^{4/3}$ distinct values of $B$, unless the curve $C$ has an exceptional form. This strengthens a result of Charalambides in several ways. The proof is based on that of Pach and De Zeeuw, who proved a similar statement for the Euclidean distance function in the real plane. Our main motivation for this paper is that for bilinear forms, this approach becomes more natural, and should better lend itself to understanding and generalization.

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