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Simultaneous Additive Equations:Repeated and Differing Degrees

Published online by Cambridge University Press:  20 November 2018

Julia Brandes and
Scott T. Parsell
Julia Brandes
Affiliation:
Mathematical Sciences, Chalmers Institute of Technology and University of Gothenburg, 412 96 Göteborg, Sweden e-mail: brjulia@chalmers.se
Scott T. Parsell
Affiliation:
Department of Mathematics, West Chester University, 25 University Ave., West Chester, PA 19383, USA e-mail: sparsell@wcupa.edu
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Abstract

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We obtain bounds for the number of variables required to establish Hasse principles, both for the existence of solutions and for asymptotic formulæ, for systems of additive equations containing forms of differing degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning–Heath–Brown, which give weak results when specialized to the diagonal situation, this is the first result on such “hybrid” systems. We also obtain specialized results for systems of quadratic and cubic forms, where we are able to take advantage of some of the stronger methods available in that setting. In particular, we achieve essentially square root cancellation for systems consisting of one cubic and $r$ quadratic equations.

Keywords

11D7211D4511P55equations in many variablescounting solutions of Diophantine equationsapplications of the Hardy–Littlewood method

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 2 , 01 April 2017, pp. 258 - 283
Copyright
Copyright © Canadian Mathematical Society 2017

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