caa a22 4500 44586866X CHVBK 20180317145505.0 cr unu---uuuuu 170323e20110901xx s 000 0 eng 10.1007/s00605-010-0248-2 doi (NATIONALLICENCE)springer-10.1007/s00605-010-0248-2 Marmon Oscar Mathematical Sciences, Chalmers University of Technology, 412 96, Gothenburg, Sweden aut Sums and differences of four k th powers [Elektronische Daten] [Oscar Marmon] We prove an upper bound for the number of representations of a positive integer N as the sum of four kth powers of integers of size at most B, using a new version of the determinant method developed by Heath-Brown, along with recent results by Salberger on the density of integral points on affine surfaces. More generally we consider representations by any integral diagonal form. The upper bound has the form $${O_{N}(B^{c/\sqrt{k}})}$$, whereas earlier versions of the determinant method would produce an exponent for B of order k −1/3 (uniformly in N) in this case. Furthermore, we prove that the number of representations of a positive integer N as a sum of four kth powers of non-negative integers is at most $${O_{\varepsilon}(N^{1/k+2/k^{3/2}+\varepsilon})}$$ for k ≥ 3, improving upon bounds by Wisdom. Springer-Verlag, 2010 Sum of k th powers nationallicence Determinant method nationallicence Diagonal form nationallicence Integral points nationallicence Monatshefte für Mathematik Springer Vienna 164/1(2011-09-01), 55-74 0026-9255 164:1<55 2011 164 605 https://doi.org/10.1007/s00605-010-0248-2 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00605-010-0248-2 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Marmon Oscar Mathematical Sciences, Chalmers University of Technology, 412 96, Gothenburg, Sweden aut NATIONALLICENCE 773 0- Monatshefte für Mathematik Springer Vienna 164/1(2011-09-01), 55-74 0026-9255 164:1<55 2011 164 605 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer