Simultaneous Additive Equations: Repeated and Differing Degrees
Artikel i vetenskaplig tidskrift, 2017
We obtain bounds for the number of variables required to establish Hasse principles, both for the existence of solutions and for asymptotic formula, 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.
applications of the Hardy-Littlewood method
equations in many variables
counting solutions of Diophantine equations