On polynomials of prescribed height in finite fields

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Abstract

This paper deals with the set m(B) of monic polynomials of degree n with integral coefficients belonging to a given n-dimensional cube B with side h. An asymptotic formula is obtained for the number of polynomials in m(B) having a specific type of decomposition into irreducible factors modulo some prime p, and an asymptotic formula for the number of primitive polynomials modulo p in m(B), which translates when n=1 into known results of I. M. Vinogradov on the distribution of primitive roots. These asymptotic formulas are nontrivial when n=1. p Moreover, an asymptotic formula is obtained for the average value of the number of divisors modulo p of polynomials in m(B), a result that is nontrivial when h ≥ max(p12/n in p, p1/2 In p.Bibliography: 11 titles.

Original languageEnglish
Pages (from-to)247-255
Number of pages9
JournalMathematics of the USSR - Sbornik
Volume63
Issue number1
DOIs
Publication statusPublished - 28 Feb 1989
Externally publishedYes

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