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This article is cited in 5 scientific papers (total in 5 papers)
On polynomials of prescribed height in finite fields
I. E. Shparlinski
Abstract:
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.
Received: 26.10.1986
Citation:
I. E. Shparlinski, “On polynomials of prescribed height in finite fields”, Mat. Sb. (N.S.), 135(177):2 (1988), 253–260; Math. USSR-Sb., 63:1 (1989), 247–255
Linking options:
https://www.mathnet.ru/eng/sm1699https://doi.org/10.1070/SM1989v063n01ABEH003271 https://www.mathnet.ru/eng/sm/v177/i2/p253
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Abstract page: | 241 | Russian version PDF: | 83 | English version PDF: | 5 | References: | 42 |
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