On the size of primitive sets in function fields

TL;DR

This paper explores primitive sets of monic polynomials over finite fields, generalizing classical results from integers, including density bounds, sum bounds, and growth rates, along with new analogues of number theory theorems.

Contribution

It introduces new bounds and generalizations for primitive polynomial sets in function fields, extending classical integer results to this setting.

Findings

Existence of primitive sets with density close to (q-1)/q

Sum over primitive sets is uniformly bounded

Generalization of Hardy-Ramanujan theorem for function fields

Abstract

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 generalize a result of Besicovitch to show that there exist primitive sets in $\mathbb{F}_q[x]$ with upper density arbitrarily close to $\frac{q - 1}{q}$. Then, for a primitive set $A$, we consider the sum $\sum_{a \in A} \frac{1}{q^{\deg a}\deg a}$, the natural analogue in this setting of a sum considered by Erd\H{o}s for primitive subsets of the integers, and show that it is uniformly bounded over all primitive sets $A$. We end with a generalization of work 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.