A Sylvester-Gallai result in the complex plane

Abstract: We show that for a Sylvester-Gallai configuration in $\mathbb{C}^2$ lying on a family of $m$ concurrent lines, each line in the family can contain at most $3m-9$ points of the set, not including the common point. This implies that many points lying on a family of concurrent lines must admit an ordinary line. We also introduce a conjecture which would improve this bound to $m-1$, which is sharp. Our approach involves ordering points by their real part, which is a new technique for studying complex line arrangements.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

Registration for the conference is free. Register at cant2021.eventbrite.com.

The conference website is www.theoryofnumbers.com/cant/ Lectures will be broadcast on Zoom. The Zoom login will be emailed daily to everyone who has registered on eventbrite. To join the meeting, you may need to download the free software from www.zoom.us.

The conference program, list of speakers, and abstracts are posted on the external website.