Coefficient growth in square chains

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.