Finite Fields and Their Applications ( IF 1 ) Pub Date : 2020-03-20 , DOI: 10.1016/j.ffa.2020.101658 Andrés Gómez-Colunga , Charlotte Kavaler , Nathan McNew , Mirilla Zhu
A set is primitive if no element of the set divides another. We consider primitive sets of monic polynomials over a finite field and find natural generalizations of many of the results known for primitive sets of integers. In particular, we show that primitive sets in the function field have lower density zero by showing that the sum , an analogue of a sum considered by Erdős, is uniformly bounded over all primitive sets A. We then adapt a method of Besicovitch to construct primitive sets in with upper density arbitrarily close to and generalize a result of Martin and Pomerance on the asymptotic growth rate of the counting function of a primitive set. Along the way we prove a quantitative analogue of the Hardy-Ramanujan theorem for function fields, as well as bounds on the size of the k-th irreducible polynomial.
中文翻译:
关于函数字段中原语集的大小
如果集合中的任何元素都不划分另一个,则该集合是原始的。我们考虑有限域上的一元多项式的原始集,并发现许多已知的整数原始集的结果的自然概括。特别地,通过显示和,我们表明函数域中的原始集具有较低的密度零。,是Erdős考虑的总和的类似物,均匀地限制在所有原始集A上。然后,我们采用Besicovitch的方法来构造 高密度任意接近 并推广了Martin和Pomerance关于原始集计数函数的渐近增长率的结果。一路上,我们证明了函数场的Hardy-Ramanujan定理的定量类似物,以及第k个不可约多项式的大小的界限。