The Szemerédi-Trotter theorem over arbitrary field of characteristic zero

Let $\mathcal{P}$ be a set of $m$ points and $\mathcal{L}$ a set of $n$ lines in $K^2$, where $K$ is a field with char$(K)=0$. We prove the incidence bound $$\mathcal{I}(\mathcal{P},\mathcal{L})=O(m^{2\mathrm{/}3}{n}^{2\mathrm{/}3}+m+n)\mathrm{.}$$ Moreover, this bound is sharp and cannot be improved. This resolves the Szemerédi-Trotter incidence problem for arbitrary field of characteristic zero. The key tool of our proof is the Baby Lefschetz principle, which allows us to reduce the problem to the complex case. Based on this observation, we further derive several related results over $K$, including Beck’s theorem, the Erdős-Szemerédi sum-product estimate, and incidence theorems involving more general algebraic objects.