Beck's Theorem for Plane Curves

Abstract

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 $$|\mathcal{D}_{d,\mathbb{K}}(A)|=\Theta_d\left(|A|^d\prod_{e=1}^d\left(\min_{C\in \mathcal{C}_{\leq e,\mathbb{K}}}|A\setminus C|\right)^{d-e+1}\right).$$