A note on multiple exponential sums in function fields
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 handling sums modulo powers of irreducibles \(w^l\) via two separate arguments according to the size of \(\langle w \rangle\), both involving induction on \(d\), and then expressing \(S(G;g)\) as a product of such sums. The author speculates that the exponent \(1/(2k)\) may be improved to \(1/k\), in correspondence with the Arkhipov-Karatsuba-Chubarikov result on the integer case, by employing different techniques.
scientific article; zbMATH DE number 6005614
scientific article