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# The Situation over Z: Results

Suppose we are given an integer monic polynomial f (x) Z[x] of degree n ≥ 2 which, over C, is “weakly supermorse,” meaning that it has n distinct roots in C, its derivative f ′ (x) has n – 1 distinct roots (the critical points) αi C, and the n – 1 values f (αi) (the critical values) are all distinct in C. Denote by S the set of critical values. Suppose that S is not equal to any nontrivial multiplicative translate aS, for any a ≠ 1 in C×. It is standard that for all but finitely many primes p, the reduction mod p of f will satisfy all the hypotheses of Theorem 17.6. Let us say such a prime p is good for f.

Choose a prime, ℓ and a field isomorphism ι : Q C. For each p ≠ which is good for f, form ...

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