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Factorization in \(\mathbb F_q(X)\). (Factorisation dans \(\mathbb F_q(X)\).) (French) Zbl 0492.12010

For given positive integers \(k\) and \(n\) let \(p_k(n)\) be the number of monic polynomials over the finite field \(\mathbb F_q\), having degree \(n\) and exactly \(k\) distinct monic irreducible factors. The author presents a different proof of an estimate due to S. D. Cohen [Proc. Edinb. Math. Soc., II. Ser. 16, 349–363 (1969; Zbl 0188.11101)], namely \[ p_k(n) = \frac{q^n(\log n)^{k-1}}{n(k-1)!} + O\left(\frac{q^n(\log n)^{k-2}}{n}\right), \] where the implied constant depends only on \(q\) and \(k\). The same estimate holds for the number of squarefree monic polynomials over \(\mathbb F_q\), having degree \(n\) and exactly \(k\) distinct monic irreducible factors and for the number of monic polynomials over \(\mathbb F_q\), having \(n\) as degree and \(k\) as the total number of monic irreducible factors (counting multiplicities).

MSC:

11T06 Polynomials over finite fields
11N45 Asymptotic results on counting functions for algebraic and topological structures

Citations:

Zbl 0188.11101
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