The Chebotarev density theorem for function fields — Incomplete intervals

Abstract

We prove a Pólya-Vinogradov type variation of the Chebotarev density theorem for function fields over finite fields valid for “incomplete intervals” I⊂Fp, provided (p1/2log⁡p)/|I|=o(1). Applications include density results for irreducible trinomials in Fp[x], i.e. the number of irreducible polynomials in the set {f(x)=xd+a1x+a0∈Fp[x]}a0∈I0,a1∈I1 is ∼|I0|⋅|I1|/d provided |I0|>p1/2+ϵ, |I1|>pϵ, or |I1|>p1/2+ϵ, |I0|>pϵ, and similarly when xd is replaced by any monic degree d polynomial in Fp[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 Sd.