Papers › A finiteness theorem for specializations of dynatomic polynomials
A finiteness theorem for specializations of dynatomic polynomials
David Krumm
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Let t and x be indeterminates, let ϕ(x)=x²+t∈ℚ(t)[x], and for every positive integer n let Φₙ(t,x) denote the nᵗʰ dynatomic polynomial of ϕ. Let Gₙ be the Galois group of Φₙ over the function field ℚ(t), and for c∈ℚ let G_(n,c) be the Galois group of the specialized polynomial Φₙ(c,x). It follows from Hilbert's irreducibility theorem that for fixed n we have Gₙ≅G_(n,c) for every c outside a thin set Eₙ⊂ℚ. By earlier work of Morton (for n=3) and the present author (for n=4), it is known that Eₙ is infinite if n≤4. In contrast, we show here that Eₙ is finite if n∈{5,6,7,9}. As an application of this result we show that, for these values of n, the following holds with at most finitely many exceptions: for every c∈ℚ, more than 81% of prime numbers p have the property that the polynomial x²+c does not have a point of period n in the p-adic field ℚₚ.
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