Kuo, WentangLiu, Yu-RuZhao, Xiaomei2023-10-032023-10-032014https://doi.org/10.4153/cjm-2013-014-9http://hdl.handle.net/10012/20003This article has been published in a revised form in the Canadian Journal of Mathematics https://doi.org/10.4153/CJM-2013-014-9. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. Copyright © Canadian Mathematical Society 2014.Let Fq[t] denote the polynomial ring over the finite field Fq. We employ Wooley’s new efficient congruencing method to prove certain multidimensional Vinogradov-type estimates in Fq[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 Fq[t].enAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Vinogradov's mean value theoremfunction fieldscircle methodArticle