Hostname: page-component-89b8bd64d-ktprf Total loading time: 0 Render date: 2026-05-07T07:49:01.453Z Has data issue: false hasContentIssue false
  • English
  • Français

RATIONAL POINTS ON CUBIC HYPERSURFACES THAT SPLIT INTO FOUR FORMS

Part of: Diophantine equations Forms and linear algebraic groups Arithmetic problems. Diophantine geometry Additive number theory; partitions

Published online by Cambridge University Press:  29 January 2016

Boqing Xue  and
Lilu Zhao
Boqing Xue
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China email boqing_xue@hotmail.com
Lilu Zhao
Affiliation:
School of Mathematics, Hefei University of Technology, Hefei 230009, China email zhaolilu@gmail.com
Get access

Abstract

Let $C\in \mathbb{Z}[x_{1},\ldots ,x_{n}]$ be a cubic form. Assume that $C$ splits into four forms. Then $C(x_{1},\ldots ,x_{n})=0$ has a non-trivial integer solution provided that $n\geqslant 10$ .

MSC classification

Primary: 11D72: Equations in many variables
Secondary: 11E76: Forms of degree higher than two 11P55: Applications of the Hardy-Littlewood method 14G25: Global ground fields

Information

Type
Research Article
Information
Mathematika , Volume 62 , Issue 2 , 2016 , pp. 430 - 440
Copyright
Copyright © University College London 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Browning, T. D., Quantitative Arithmetic of Projective Varieties (Progress in Mathematics 277 ), Birkhäuser (2009).Google Scholar
Browning, T. D., Rational points on cubic hypersurfaces that split off a form. With an appendix by J.-L. Colliot-Thélène. Compos. Math. 146 2010, 853–885.Google Scholar
Browning, T. D., A survey of applications of the circle method to rational points. In Arithmetic and Geometry (London Mathematical Society Lecture Note Series 420 ), Cambridge University Press (2015), 89–113.Google Scholar
Browning, T. D. and Heath-Brown, D. R., Integral points on cubic hypersurfaces. In Analytic Number Theory: Essays in Honour of Klaus Roth, Cambridge University Press (Cambridge, 2009), 75–90.Google Scholar
Brüdern, J. and Wooley, T. D., On Waring’s problem for cubes and smooth Weyl sums. Proc. Lond. Math. Soc. (3) 82 2001, 89–109.Google Scholar
Colliot-Thélène, J. L. and Salberger, P., Arithmetic on some singular cubic hypersurfaces. Proc. Lond. Math. Soc. 58 1989, 519–549.Google Scholar
Davenport, H., Cubic forms in thirty-two variables. Philos. Trans. R. Soc. Lond. A 251 1959, 193–232.Google Scholar
Davenport, H., Analytic Methods for Diophantine Equations and Diophantine Inequalities, 2nd edn., (ed. Browning, T. D.), Cambridge University Press (Cambridge, 2005).Google Scholar
Heath-Brown, D. R., Cubic forms in ten variables. Proc. Lond. Math. Soc. 47 1983, 225–257.CrossRefGoogle Scholar
Heath-Brown, D. R., Cubic forms in 14 variables. Invent. Math. 170 2007, 199–230.CrossRefGoogle Scholar
Hooley, C., On nonary cubic forms. J. reine angew. Math. 386 1988, 32–98.Google Scholar
Iwaniec, H. and Kowalski, E., Analytic Number Theory (Colloquium Publications 53 ), American Mathematical Society (Providence, RI, 2004).Google Scholar
Mordell, L. J., A remark on indeterminate forms in several variables. J. Lond. Math. Soc. 12 1937, 127–129.Google Scholar
Vaughan, R. C., A new iterative method in Waring’s problem. Acta Math. 162 1989, 1–71.Google Scholar
Vaughan, R. C., The Hardy–Littlewood Method, 2nd edn., Cambridge University Press (Cambridge, 1997).Google Scholar
Wooley, T. D., Sums of three cubes. Mathematika 47 2000, 53–61.Google Scholar
Xue, B. and Dai, H., Rational points on cubic hypersurfaces that split off two forms. Bull. Lond. Math. Soc. 46 2014, 169–184.Google Scholar