Unlikely intersections for two-parameter families of polynomials

Joint IAS/Princeton University Number Theory Seminar
Topic:Unlikely intersections for two-parameter families of polynomials
Speaker:Tom Tucker
Affiliation:Rochester University
Date:Thursday, October 22
Time/Room:4:30pm - 5:30pm/Fine 214, Princeton University

Inspired by work of Masser and Zannier for torsion specializations of points on the Legendre elliptic curve, Baker and DeMarco proved that if $v$, $w$ are two points in $C$, then there are at most finitely many $t$ in $C$ such that $v$ and $w$ are both preperiodic for the polynomial $x^2 + t$, unless of course $v$ equals plus or minus $w$. Here we prove a two-dimensional version of this result, namely that if $v$, $w$, and $z$ are distinct complex numbers, then the set of parameters $(a, b)$ such that $v$, $w$, and $z$ are all preperiodic under $f(x) = x^3 + ax + b$ cannot be Zariski dense in the affine plane. This represents joint work with Liang-Chung Hsia and Dragos Ghioca.