부산대학교 도서관

Let $f_r$ denote the quadratic polynomial $f_r(x) = x^2+r$. The paperstudies the number $I_{r}(n)$ of irreducible factors of the iterates$f_r^{\circ n}$ of $f_r$. Polynomials for which $I_{r}(n)$ is boundedindependently of $n$ are called {\it eventually stable}. It is an openproblem to show that if 0 has an infinite orbit under $f_r$, then $f_r$is eventually stable. This question is related to the theory ofarboreal representations. \parS. Hamblen, R. Jones and K. Madhu [Int. Math. Res. Not. IMRN {\bf 2015},no.~7, 1924--1958; MR3335237] established eventual stabilityfor all quadratic polynomials $f_r$ except for the case $r=1/c$, $c \in\Bbb{Z}\setminus\{0,-1\}$. The paper under review addresses the remaining cases.Avariety of conditions on $c$ are established that imply eventualstability of $f_{1/c}$. For example, eventual stability is proved inthe following cases:\roster\item $c<0$ and $-c$ is not a square;\item $c>0$ and $c$ is odd;\item $c+1$ and $-c$ are not squares and $c+1$ has a prime divisor congruentto3 modulo 4.\item $c$ is not of the form $4m^2(m^2-1)$ and $|c| \leqslant 10^{1000}$.\endroster Some results are obtained for bounding $I_{1/c}(n)$. In particular,for $|c| \leqslant 10^9$ the polynomials $f_{1/c}$ are eventuallystable and $I_{{1/c}}(n) \leqslant 4$. The paper makes a preciseconjecture, supported by computation, for the exact values of$I_{1/c}(n)$. The conjecture implies that $I_{1/c}(n) \leqslant 4$ forall $c \in \Bbb{Z} \setminus\{0, -1\}$ and all $n$ and that for adensity one set of integers ${c \in \Bbb{Z}\setminus \{0, -1\}}$, one has$I_{1/c}(n)=1$ for all $n$.