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# Multidimensional Vinogradov-type Estimates in Function Fields

Published:2013-05-07
Printed: Aug 2014
• Wentang Kuo,
Department of Pure Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, ON N2L 3G1
• Yu-Ru Liu,
Department of Pure Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, ON N2L 3G1
• Xiaomei Zhao,
Department of Mathematics, Central China Normal University, Wuhan, Hubei, China 430079
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## Abstract

Let $\mathbb{F}_q[t]$ denote the polynomial ring over the finite field $\mathbb{F}_q$. We employ Wooley's new efficient congruencing method to prove certain multidimensional Vinogradov-type estimates in $\mathbb{F}_q[t]$. These results allow us to apply a variant of the circle method to obtain asymptotic formulas for a system connected to the problem about linear spaces lying on hypersurfaces defined over $\mathbb{F}_q[t]$.
 Keywords: Vinogradov's mean value theorem, function fields, circle method
 MSC Classifications: 11D45 - Counting solutions of Diophantine equations 11P55 - Applications of the Hardy-Littlewood method [See also 11D85] 11T55 - Arithmetic theory of polynomial rings over finite fields