On the Sylvester–Gallai theorem for conics

Czapliński, Adam; Dumnicki, Marcin; Farnik, Łucja; Gwoździewicz, Janusz; Lampa-Baczyńska, Magdalena; Malara, Grzegorz; Szemberg, Tomasz; Szpond, Justyna; Tutaj-Gasińska, Halszka

doi:10.4171/rsmup/136-13

Cited by 8 publications

(7 citation statements)

References 12 publications

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“…In fact Green and Tao give a complete classification of the sets of n points S spanning less than cn(log log n) c ordinary lines. Their paper inspired not only [1] but also articles concerning ordinary circles [8] and ordinary conics [4] and [5].…”

Section: Introductionmentioning

confidence: 99%

On Sets Defining Few Ordinary Planes

Ball

2017

Discrete Comput Geom

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Let S be a set of n points in real four-dimensional space, no four coplanar and spanning the whole space. We prove that if the number of solids incident with exactly four points of S is less than Kn 3 for some K = o(n 1 7 ) then, for n sufficiently large, all but at most O(K) points of S are contained in the intersection of five linearly independent quadrics. Conversely, we prove that there are finite subgroups of size n of an elliptic curve which span less than 1 6 n 3 solids containing exactly four points of S.

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Section: Introductionmentioning

confidence: 99%

On Sets Defining Few Ordinary Planes

Ball

2017

Discrete Comput Geom

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“…The proof of Wiseman and Wilson was somewhat lengthy and convoluted. Czaplinski et al [3] provided a shorter proof using Cremona transformations. Another short proof was given by Boys, Valculescu, and De Zeeuw [1], based on Veronese mappings.…”

Section: Theorem 12 (Wiseman-wilson)mentioning

confidence: 99%

“…Note that a curve of degree d is typically determined by d(d + 3)/2 points, so the statement should be as follows: If a finite set of points in R 2 is not contained in a curve of degree d, then there is a curve of degree d that contains exactly d(d + 3)/2 of these points. The papers [3,1] both mention this conjecture, but their techniques appear to be difficult to extend to curves of higher degree.…”

Section: Theorem 12 (Wiseman-wilson)mentioning

confidence: 99%

“…It should also be noted that in our Theorem 1.3, the cubic is not necessarily determined by the nine points, i.e., it may not be the only cubic containing those nine points. In each of [8,3,1], the conic containing five points is also determined by those five points. Our proof does not seem to be able to give a cubic through nine points that is also determined by the nine points.…”

Section: Theorem 12 (Wiseman-wilson)mentioning

confidence: 99%

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A Sylvester-Gallai theorem for cubic curves

Cohen¹,

Zeeuw²

2020

Preprint

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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 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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“…Beck's theorem has important applications in different areas of mathematics, and it has opened a new research field in combinatorial geometry, see for instance [1], [4], [6], [7], [13], [16]. Another important family of problems in combinatorial geometry is to bound the number of curves with a given degree that are determined by A and satisfy other conditions (for example, in the Sylvester-Gallai type results, the curves have to pass through few points of A), see for instance [2], [3], [5], [19]. Thus it seems natural and important to ask if Beck's theorem can be generalized for conics, cubics, etc.…”

Section: Introductionmentioning

confidence: 99%

Beck's Theorem for Plane Curves

Huicochea¹

2020

Electron. J. Combin.

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Let $d\in\mathbb{Z}^+t$, $\mathbb{K}$ be a field of characteristic zero and $A$ be a nonempty finite subset of $\mathbb{K}^2$. Denote by $\mathcal{C}_{d,\mathbb{K}}$ the family of algebraic curves of degree $d$ in $\mathbb{K}^2$ and $\mathcal{C}_{\leq d,\mathbb{K}}:=\bigcup_{e=1}^d\mathcal{C}_{e,\mathbb{K}}$. For any $C_1\in \mathcal{C}_{d,\mathbb{K}}$, we say that $C_1$ is determined by $A$ if for any $C_2\in\mathcal{C}{d,\mathbb{K}}$ such that $C_2\cap A\supseteq C_1\cap A$, we have that $C_1=C_2$; we denote by $\mathcal{D}_{d,\mathbb{K}}(A)$ the family of elements of $\mathcal{C}_{d,\mathbb{K}}$ determined by $A$. Beck's theorem establishes that if $\mathbb{K}=\mathbb{R}$ and $A$ is not collinear, then $$|\mathcal{D}_{1,\mathbb{R}}(A)|=\Theta\left(|A|\min_{C\in \mathcal{C}_{1,\mathbb{R}}}|A\setminus C|\right).$$ In this paper we generalize Beck's theorem showing that for all $d\in\mathbb{Z}^+$, there exists a constant $c=c(d)>0$ such that if $\min_{C\in\mathcal{C}_{\leq d,\mathbb{K}}}|A\setminus C|>c,$ then $$|\mathcal{D}_{d,\mathbb{K}}(A)|=\Theta_d\left(|A|^d\prod_{e=1}^d\left(\min_{C\in \mathcal{C}_{\leq e,\mathbb{K}}}|A\setminus C|\right)^{d-e+1}\right).$$

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On the Sylvester–Gallai theorem for conics

Cited by 8 publications

References 12 publications

On Sets Defining Few Ordinary Planes

On Sets Defining Few Ordinary Planes

A Sylvester-Gallai theorem for cubic curves

Beck's Theorem for Plane Curves

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