The Chebotarev density theorem for function fields -- incomplete intervals

2019/01/20 by Pär Kurlberg, Kurlberg, Pär, Lior Rosenzweig +1
Mathematics · #11N05 #11T06 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11N05 #msc:11T06

paper · pdf · doi:10.48550/arxiv.1901.06751

Minor revision

arxiv created 2020/07/04 · arxiv updated 2020/07/07

Abstract

We prove a Polya-Vinogradov type variation of the the Chebotarev density theorem for function fields over finite fields valid for "incomplete intervals" I ⊂ \mathbbF_p, provided (p^1/2log p)/|I| = o(1). Applications include density results for irreducible trinomials in \mathbbF_p[x], i.e. the number of irreducible polynomials in the set \ f(x) = x^d + a₁ x + a₀ ∈ \mathbbF_p[x] \_a₀ ∈ I₀, a₁∈ I₁ is ∼ |I₀|⋅ |I₁|/d provided |I₀| > p^1/2+ε, |I₁| > p^ε, or |I₁| > p^1/2+ε, |I₀| > p^ε, and similarly when x^d is replaced by any monic degree d polynomial in \mathbbF_p[x]. Under the above assumptions we can also determine the distribution of factorization types, and find it to be consistent with the distribution of cycle types of permutations in the symmetric group S_d.

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