Skip to main content
×
×
Home

VINOGRADOV SYSTEMS WITH A SLICE OFF

  • Julia Brandes (a1) (a2) (a3) and Trevor D. Wooley (a4)
Abstract

Let $I_{s,k,r}(X)$ denote the number of integral solutions of the modified Vinogradov system of equations

$$\begin{eqnarray}x_{1}^{j}+\cdots +x_{s}^{j}=y_{1}^{j}+\cdots +y_{s}^{j}\quad (1\leqslant j\leqslant k,\;j\neq r),\end{eqnarray}$$
with $1\leqslant x_{i},y_{i}\leqslant X\;(1\leqslant i\leqslant s)$ . By exploiting sharp estimates for an auxiliary mean value, we obtain bounds for $I_{s,k,r}(X)$ for $1\leqslant r\leqslant k-1$ . In particular, when $s,k\in \mathbb{N}$ satisfy $k\geqslant 3$ and $1\leqslant s\leqslant (k^{2}-1)/2$ , we establish the essentially diagonal behaviour $I_{s,k,1}(X)\ll X^{s+\unicode[STIX]{x1D700}}$ .

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      VINOGRADOV SYSTEMS WITH A SLICE OFF
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      VINOGRADOV SYSTEMS WITH A SLICE OFF
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      VINOGRADOV SYSTEMS WITH A SLICE OFF
      Available formats
      ×
Copyright
This article is distributed with Open Access under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided that the original work is properly cited.
References
Hide All
1. Bourgain, J., Demeter, C. and Guth, L., Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. Ann. of Math. (2) 184(2) 2016, 633682.
2. Brandes, J. and Parsell, S. T., Simultaneous additive equations: repeated and differing degrees. Canad. J. Math. 69(2) 2017, 258283.
3. Brüdern, J. and Robert, O., A paucity estimate related to Newton sums of odd degree. Mathematika 58(2) 2012, 225235.
4. Brüdern, J. and Robert, O., Rational points on linear slices of diagonal hypersurfaces. Nagoya Math. J. 218 2015, 51100.
5. Macdonald, I. G., Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs (Oxford, 1979).
6. Parsell, S. T. and Wooley, T. D., A quasi-paucity problem. Michigan Math. J. 50 2002, 461469.
7. Wooley, T. D., A note on symmetric diagonal equations. In Number Theory with an Emphasis on the Markoff Spectrum (Provo, UT, 1991) (eds Pollington, A. D. and Moran, W.), Dekker (New York, 1993), 317321.
8. Wooley, T. D., Rational solutions of pairs of diagonal equations, one cubic and one quadratic. Proc. Lond. Math. Soc. (3) 110(2) 2015, 325356.
9. Wooley, T. D., The cubic case of the main conjecture in Vinogradov’s mean value theorem. Adv. Math. 294 2016, 532561.
10. Wooley, T. D., Nested efficient congruencing and relatives of Vinogradov’s mean value theorem. Preprint, 2017, arXiv:1708.01220.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

MSC classification