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• W2100395279 abstract "We examine the question of when a quadratic polynomial f(x) defined over a number field K can have a newly reducible nth iterate, that is, f^n(x) irreducible over K but f^{n+1}(x) reducible over K, where f^n denotes the nth iterate of f. For each choice of critical point gamma of f(x), we consider the family g_{gamma,m}(x)= (x - gamma)^2 + m + gamma, m in K. For fixed n geq 3 and nearly all values of gamma, we show that there are only finitely many m such that g_{gamma,m} has a newly reducible nth iterate. For n = 2 we show a similar result for a much more restricted set of gamma. These results complement those obtained by Danielson and Fein in the higher-degree case. Our method involves translating the problem to one of finding rational points on certain hyperelliptic curves, determining the genus of these curves, and applying Faltings' theorem." @default.
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• W2100395279 date "2012-12-31" @default.
• W2100395279 modified "2023-09-27" @default.
• W2100395279 title "Newly reducible iterates in families of quadratic polynomials" @default.
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• W2100395279 doi "https://doi.org/10.2140/involve.2012.5.481" @default.
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