Motivated by coding applications,two enumeration problems are considered: the number of distinct divisors of a degree-m polynomial over F = GF(q), and the number of ways a polynomial can be written as a product of two polynomials of degree at most n over F. For the two problems, bounds are obtained on the maximum number of factorizations, and a characterization is presented for polynomials attaining that maximum. Finally, expressions are presented for the average and the variance of the number of factorizations, for any given m (respectively, n).

中文翻译：

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关于有限域上多项式的因式分解次数

受编码应用的启发，考虑了两个枚举问题：F = GF(q) 上的 m 次多项式的不同因数的数量，以及多项式最多可以写成两个次多项式的乘积的方式数n 超过 F。对于这两个问题，获得了最大分解次数的界限，并给出了达到该最大值的多项式的特征。最后，对于任何给定的 m（分别为 n），给出了分解次数的平均值和方差的表达式。