A proof of the Schinzel-Zassenhaus conjecture on polynomials

2019/12/28 by Dimitrov, Vesselin · 2 citations
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1912.12545

Abstract

We prove that if P(X) ∈ ℤ[X] is an integer polynomial of degree n and having P(0) = 1, then either P(X) is a product of cyclotomic polynomials, or else at least one of the complex roots of P belongs to the disk |z| ≤ 2 ^{- 1 / (4n)} . We also obtain a relative version of this result over the compositum ℚ^ab ⋅ ℚ^t.p of all abelian and all totally p-adic extensions of ℚ, for any fixed prime~p, and apply it to prove a ℚ^ab ⋅ ℚ^t.p-relative canonical height lower bound on the multiplicative group. Another extension is given to a uniform positive height lower bound, inverse-proportional to the total number of singular points, on holonomic power series in ℚ[[X]] and not of the form p(X) / (X^k-1)^m, where p(X) ∈ ℚ[X], with a further application to existence of a small critical value for certain rational functions.

Cited by

• The exceptional set in Cassel's theorem on small cyclotomic integers
• On the irreducibility of the non-cyclotomic part of most 0,1-polynomials with few terms

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• The Schinzel Hypothesis for Polynomials
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