# Coefficient growth in square chains

## TL;DR

This paper investigates the growth of coefficients in iterated quadratic polynomials that factor over integers, showing they grow exponentially and become too large for explicit construction as the number of iterations increases.

## Contribution

It establishes new bounds on prime divisors and exponential growth of coefficients in square chains, highlighting limitations in constructing such polynomials explicitly.

## Key findings

- Prime divisors of coefficients are bounded by powers of 2.
- Coefficients grow exponentially, exceeding manageable sizes for large chains.
- Growth estimates suggest explicit construction of large chains is infeasible.

## Abstract

Suppose $((\cdots((x^{2}-c_{1})^{2}-c_{2})^{2}\cdots)^{2}-c_{k-1})^{2}-c_{k}$ splits into linear factors over $\mathbb{Z}$ and $c_{k}\neq0$. We show that for each $j$ and each prime $p$, if $p\leq2^{j-1}$ then $p$ divides $c_{j}$. Consequently, $$\ln c_{j}>\frac{1}{4}\cdot2^{j}\,\,\mathrm{for}\,j\geq5$$ If we also have $p\equiv3\,(\mathrm{mod\,4)}$ then $p^{2^{j-\left\lceil \lg p\right\rceil }}$ divides $c_{j}$. Consequently, if $k\geq3$, there exists some absolute constant $\lambda>0$ so that, $$\ln c_{j}>\lambda k2^{j}\mathrm{\,\,for\,all\,}j$$ These estimates argue against the possibility of explicitly constructing polynomials of the given form for large $k$, as the coefficients quickly become too large to manipulate.