부산대학교 도서관

Let $f_r$ denote the quadratic polynomial $f_r(x) = x^2+r$. The paper studies 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 bounded independently of $n$ are called {\it eventually stable}. It is an open problem 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 of arboreal representations. \par S. Hamblen, R. Jones and K. Madhu [Int. Math. Res. Not. IMRN {\bf 2015}, no.~7, 1924--1958; MR3335237] established eventual stability for 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. A variety of conditions on $c$ are established that imply eventual stability of $f_{1/c}$. For example, eventual stability is proved in the 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 congruent to 3 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 eventually stable and $I_{{1/c}}(n) \leqslant 4$. The paper makes a precise conjecture, supported by computation, for the exact values of $I_{1/c}(n)$. The conjecture implies that $I_{1/c}(n) \leqslant 4$ for all $c \in \Bbb{Z} \setminus\{0, -1\}$ and all $n$ and that for a density one set of integers ${c \in \Bbb{Z}\setminus \{0, -1\}}$, one has $I_{1/c}(n)=1$ for all $n$.