关于有限域上多项式分解的个数
On the Number of Factorizations of Polynomials over Finite Fields
摘要：
受编码应用的启发，考虑了两个枚举问题：F = GF（q）上的m次多项式的不同因子的数量，以及多项式可以被写为F上至多n次的两个多项式的乘积的方式的数量。对于这两个问题，边界上的最大数量的因式分解，并提出了一个表征多项式达到最大。最后，表达式的平均值和方差的因子分解的数量，对于任何给定的m（分别为，n）。
英文摘要：
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).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
