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On polynomials of prescribed height in finite fields. (English. Russian original) Zbl 0665.12018
Math. USSR, Sb. 63, No. 1, 247-255 (1989); translation from Mat. Sb., Nov. Ser. 135(177), No. 2, 253-260 (1988).
Let B be a set of integer points in the n-dimensional cube with side h and integer vertices \(B=\{a_ 0,...,a_{n-1})\in {\mathbb{Z}}^ n\), \(\ell_ i\leq a_ i\leq \ell_ i+h\), \(i=0,1,...,n-1\}\), where h is a natural number, \(\ell_ 0,...,\ell_{n-1}\) are integers, M(B) is a set of the monic polynomials of degree n with coefficients from B: \(M(B)=\{x^ n+a_{n-1}x^{n-1}+...+a_ 0\), \((a_ 0,...,a_{n-1})\in B\}\). Denote by \(T_ n\) a set of all n-tuples collections \(t=(t_ 1,...,t_ n)\) of non-negative integers under the condition \(t_ 1+2t_ 2+...+nt_ n=n\). The number \(t\in T_ n\) is called a decomposition type of a polynomial \(f\in {\mathbb{F}}_ p[x]\) (p is a prime, \({\mathbb{F}}_ p\) is a finite field) if \[ f(x)=\prod^{n}_{m=1}\prod^{n}_{k=1}f_{m,k}(x), \] where \(f_{m,k}\) are irreducible polynomials of degree m over the field \({\mathbb{F}}_ p\), \(m=1,...,n\), \(k=1,...,t_ m.\)
It is obtained an asymptotic formula for the number of polynomials (from M(B)) whose reduction modulo p has a given decomposition type \(t\in T_ n\). For any \(\epsilon >0\) it is nontrivial for \(h\geq p^{n/n+1+\epsilon}\). This estimate is applied to a study of the decomposition types of the prime p into prime ideals in fields of algebraic numbers of a certain form.
Furthermore, an asymptotic formula for the number of polynomials \(primitive\quad mod p\) from M(B) and an estimation of the mean value for the number of \(divisors\quad mod\quad p\) of polynomials from M(B) are obtained.
Reviewer: S.Kotov

11T06 Polynomials over finite fields
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