Alice and Bob are given $n$-bit integer pairs $(x,y)$ and $(a,b)$, respectively, and they must decide if $y=ax+b$. We prove that the randomised communication complexity of this Point--Line Incidence problem is $Θ(\log n)$. This confirms a conjecture of Cheung, Hatami, Hosseini, and Shirley (CCC 2023) that the complexity is super-constant, and gives the first example of a communication problem with constant support-rank but super-constant randomised complexity.

翻译：Alice和Bob分别获得$n$位整数对$(x,y)$和$(a,b)$，他们需要判定是否满足$y=ax+b$。我们证明该点-线关联问题的随机化通信复杂度为$Θ(\log n)$。这证实了Cheung、Hatami、Hosseini和Shirley（CCC 2023）关于复杂度为超常数的猜想，并首次给出一个支持秩为常数但随机化复杂度为超常数的通信问题实例。