Eventually stable quadratic polynomials over Q

We study the number of irreducible factors (over Q) of the nth iterate of a polynomial of the form f(r)(x) = x(2) + r for r is an element of Q. When the number of such factors is bounded independent of n, we call f(r)(x) eventually stable (over Q). Previous work of Hamblen, Jones, and Madhu [8] shows that f(r) is eventually stable unless r has the form 1/c for some c is an element of Z\ {0, -1}, in which case existing methods break down. We study this family, and prove that several conditions on c of various flavors imply that all iterates of f(1/c) are irreducible. We give an algorithm that checks the latter property for all c up to a large bound B in time polynomial in log B. We find all c-values for which the third iterate of f(1/c) has at least four irreducible factors, and all c-values such that f(1/c) is irreducible but its third iterate has at least three irreducible factors. This last result requires finding all rational points on a genus-2 hyperelliptic curve for which the method of Chabauty and Coleman does not apply; we use the more recent variant known as elliptic Chabauty. Finally, we apply all these results to completely determine the number of irreducible factors of any iterate of f(1/c), for all c with absolute value at most 10(9).