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Direct factors of polynomial rings over finite fields. (English) Zbl 0591.12020

An asymptotic formula is derived for the total number of polynomials of degree n in an arbitrary direct factor of the set G of all monic polynomials in one unknown over a finite field with q elements. A direct factor of G is a subset \(B_ 1\) of G such that for some subset \(B_ 2\) of G, every polynomial w in G has a unique factorization of the form \(w=b_ 1b_ 2\), where \(b_ i\in B_ i\). The asymptotic formula is given as \(c_ 1q^ n\), where \(c_ 1\) is a constant depending on \(B_ 1\).
Reviewer: J.Liang

MSC:

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

[1] Daboussi, H., On the density of direct factors of the set of positive integers, J. London Math. Soc. (2), 19, 21-24 (1979) · Zbl 0409.10044
[2] Hardy, G. H., Divergent Series (1947), Oxford Univ. Press: Oxford Univ. Press London/New York · Zbl 0897.01044
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