On Artin's conjecture over function fields
type: Item
instance of: scholarly article
title: On Artin's conjecture over function fields
zbMATH Open document ID: 0837.11063
author: Francesco Pappalardi
author: Igor E. Shparlinski
published in: Finite Fields and their Applications
publication date: 1996-05-20T00:00:00Z
review text: Let \(C\) be a nonsingular irreducible curve of degree \(d\) over a finite field \(\mathbb{F}_q\) and let \(K\), \(\overline K\) be the corresponding function fields over \(\mathbb{F}_q\) resp. over its algebraic closure. Let \(n\) be an integer and assume that \(r \in K\) is a rational function of degree \(m\) which for no divisor \(k\) of \(q^n - 1\) is the \(k\)-th power of a rational function in \(\overline K\). Then the number of prime divisors of \(C\) of degree \(n\) with respect to which \(r\) is a primitive root equals \(\varphi (q^n - 1)/n +\) error term, which in an explicit way depends on \(m,n,d,q\) and the number of distinct prime divisors of \(q^n - 1\). The novelty of this result lies in the error term, since asymptotics in this problem have been found already by \textit{H. Bilharz} [Math. Ann. 114, 476-492 (1937; Zbl 0016.34301)] who did not restrict himself to the case of rational \(r\). The proof is easily reduced to the count of primitive roots of \(\mathbb{F}_{q^n}\) having a special form and this is done using a bound for character sums obtained by \textit{G. I. Perelmuter} [Mat. Zametki, 5, 373-380 (1969; Zbl 0179.49903)] as a consequence of Riemann's hypothesis for function fields.
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 11G20
zbMATH DE Number: 828728
zbMATH Keywords: irreducible curve
zbMATH Keywords: rational function
zbMATH Keywords: number of prime divisors
Wikidata QID: Q123268766
reviewed by: Władysław Narkiewicz
MaRDI profile type: Publication
full work available at URL: ffta.1995.1030
OpenAlex ID: W2036820083
DOI: 10.1006/FFTA.1995.1030
description: scientific article; zbMATH DE number 828728