# Simultaneously preperiodic integers for quadratic polynomials

## TL;DR

This paper characterizes the set of parameters for quadratic polynomials where two given integers are simultaneously preperiodic, extending previous results and completing the classification for certain integer pairs.

## Contribution

It completes the classification of parameters where two integers are simultaneously preperiodic for quadratic polynomials, building on Buff's approach.

## Key findings

- Set of parameters for given integers is fully described when their absolute values differ.
- Confirmed that the set is finite and explicitly determined for specific integer pairs.
- Extended previous results to a broader class of integer parameters.

## Abstract

In this article, we study the set of parameters $c \in \mathbb{C}$ for which two given complex numbers $a$ and $b$ are simultaneously preperiodic for the quadratic polynomial $f_{c}(z) = z^{2} +c$. Combining complex-analytic and arithmetic arguments, Baker and DeMarco showed that this set of parameters is infinite if and only if $a^{2} = b^{2}$. Recently, Buff answered a question of theirs, proving that the set of parameters $c \in \mathbb{C}$ for which both $0$ and $1$ are preperiodic for $f_{c}$ is equal to $\lbrace -2, -1, 0 \rbrace$. Following his approach, we complete the description of these sets when $a$ and $b$ are two given integers with $\lvert a \rvert \neq \lvert b \rvert$.