A note on multiple exponential sums in function fields
type: Item
instance of: scholarly article
title: A note on multiple exponential sums in function fields
zbMATH Open document ID: 1300.11128
author: Xiaomei Zhao
published in: Finite Fields and their Applications
publication date: 2012-02-11T00:00:00Z
review text: Suppose that the finite field \({\mathbb F}_q\) has characteristic \(p\), where \(p\) does not divide \(k\), and let \({\mathbb A} = {\mathbb F}_q[t]\). The strong analogy between \({\mathbb A}\) and \({\mathbb Z}\) leads one to consider \({\mathbb A}\)-analogues of familiar diophantine problems involving \(k\)th powers, and a modern version of the circle method appropriate to this setting has been developed recently by \textit{Y.-R. Liu} and \textit{T. D. Wooley} [J. Reine Angew. Math. 638, 1--67 (2010; Zbl 1221.11203)]. Problems in which one seeks a linear space of solutions to a diophantine problem require the investigation of multidimensional exponential sums, which have been analyzed in the integer case by \textit{G. I. Arkhipov} et al. [Tr. Mat. Inst. Steklova 151, 128 p. (1980; Zbl 0441.10037)] and more recently by the reviewer [Proc. Lond. Math. Soc. (3) 91, No. 1, 1--32 (2005; Zbl 1119.11024)]. Let \(G({\mathbf x})\) be a form of degree \(k\) in \(d\) variables, with coefficients in \({\mathbb A}\). When \(\alpha = \sum_{i \leqslant n} a_i t^i \in {\mathbb F}_q((1/t))\), define \(e(\alpha) = \exp(2\pi i \, {\text{tr}}(a_{-1})/p)\), where tr denotes the usual trace map from \({\mathbb F}_q\) to \({\mathbb F}_p\), and write \(\langle \alpha \rangle = q^n\). The author shows that if \(g\) is monic and does not share a common factor with all the coefficients of \(G\), then one has \[ S(G;g)=\sum_{{\mathbf x} \in ({\mathbb A}/(g))^d} e(G({\mathbf x})/g) \ll_{k,d,\varepsilon} \langle g \rangle^{d-\frac{1}{2k}+\varepsilon}, \] and in fact that the \(\ll_{k,d,\varepsilon} \langle g \rangle^{\varepsilon}\) can be replaced by an explicit bound in terms of divisor functions. This is the first result on multidimensional sums in the function field case and is employed in the author's subsequent work [Proc. Lond. Math. Soc. (3) 104, No. 2, 287--322 (2012; Zbl 1276.11164)] to handle the major arcs with denominator \(g\) in the circle method. The argument proceeds by first…
reviewed by: Scott T. Parsell
Mathematics Subject Classification ID: 11T23
Mathematics Subject Classification ID: 11T55
zbMATH DE Number: 6005614
zbMATH Keywords: function fields
zbMATH Keywords: exponential sums
MaRDI profile type: Publication
full work available at URL: j.ffa.2011.06.003
OpenAlex ID: W2032962521
cites work: Q3882539
cites work: Q5577214
cites work: The density of rational lines on cubic hypersurfaces
cites work: Multiple exponential sums over smooth numbers
cites work: A generalization of Vinogradov's mean value theorem
cites work: A note on simultaneous congruences
DOI: 10.1016/J.FFA.2011.06.003
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description: scientific article; zbMATH DE number 6005614