The Chebotarev density theorem for function fields-Incomplete intervals

We prove a Polya-Vinogradov type variation of the Chebotarev density theorem for function fields over finite fields valid for "incomplete intervals" I subset of F-p, provided (p(1/2) log p)/|I| = o(1). Applications include density results for irreducible trinomials in F-p[x], i.e. the number of irreducible polynomials in the set {f(x) = x(d) + a(1)x + a(0) is an element of F-p[x]}a(0) is an element of I-0,I- a(1) is an element of I-1 is similar to |I-0|.|I-1|/d provided |I-0| > p(1/2+is an element of), |I-1| > p(is an element of), or |I-1| > p(1/2+is an element of), |I-0| > p(c), and similarly when x(d) is replaced by any monic degree d polynomial in F-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. (C) 2021 Published by Elsevier Inc.