On polynomials of prescribed height in finite fields

I. E. Shparlinski

Sbornik. Mathematics, Tome 63 (1989) no. 1, pp. 247-255

Résumé

This paper deals with the set $\mathfrak M(B)$ 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 $\mathfrak M(B)$ 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 $\mathfrak M(B)$, which translates when $n=1$ into known results of I. M. Vinogradov on the distribution of primitive roots. These asymptotic formulas are nontrivial when $h\geqslant p^{n/(n+1)+\varepsilon}$ for any $\varepsilon>0$. Moreover, an asymptotic formula is obtained for the average value of the number of divisors modulo $p$ of polynomials in $\mathfrak M(B)$, a result that is nontrivial when $h\geqslant\max(p^{1-2/n}\ln p,p^{1/2}\ln p)$. Bibliography: 11 titles.