Frontiers of Mathematics in China >

2014 , Vol. 9 >Issue 3: 641 - 657

DOI: https://doi.org/10.1007/s11464-013-0334-6

RESEARCH ARTICLE

Diophantine inequality involving binary forms

  • Boqing XUE
Expand
  • Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

Received date: 16 Jul 2013

Accepted date: 24 Sep 2013

Published date: 24 Jun 2014

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
Fold

Abstract

Let r= 2d-1 + 1. We investigate the diophantine inequality

|∑i=1rλiΦi(xi,yi)+η||<(max⁡1≤i≤r{|xi|,|yi|})-σ
where Φi(x, y) ∈Z[x, y] (1≤i≤r) are nondegenerate forms of degree d= 3 or 4.

Key words： Diophantine inequality; binary form

Cite this article

Boqing XUE . Diophantine inequality involving binary forms[J]. Frontiers of Mathematics in China, 2014 , 9(3) : 641 -657 . DOI: 10.1007/s11464-013-0334-6

References

Publishing order | Descend order by publishing year | Descend order by cited within
1
BakerR C. Cubic diophantine inequalities. Mathematika, 1982, 29: 83-92

DOI

2
BakerR C, BrüdernJ, WooleyT D. Cubic diophantine inequalities. Mathematika, 1995, 42: 264-277

DOI

3
BrowningT D. Quantitative Arithmetic of Projective Varieties. Progress in Math, Vol 277. Basel: Birkhäuser, 2009

DOI

4
BrowningT D, DietmannR, ElliottP D T A. Least zero of a cubic form. Math Ann, 2012, 352: 745-778

DOI

5
BrüdernJ. Cubic diophantine inequalities. Mathematika, 1988, 35: 51-58

DOI

6
BrüdernJ. Cubic diophantine inequalities (II). J Lond Math Soc, 1996, 53(2): 1-18

DOI

7
BrüdernJ. Cubic diophantine inequalities (III). Periodica Mathematica Hungarica, 2001, 42(1-2): 211-226

DOI

8
CookR J. The value of additive forms at prime arguments. J Théor Nombres Bordeaux, 2001, 13: 77-91

DOI

9
DavenportH. Analytic Methods for Diophantine Equations and Diophantine Inequalities. 2nd ed. Cambridge: Cambridge University Press, 2005

DOI

10
DavenportH, HeilbronnH. On indefinite quadratic forms in five variables. J Lond Math Soc, 1946, 21: 185-193

DOI

11
HarveyM P. Cubic diophantine inequalities involving a norm form. Int J Number Theory, 2011, 7(8): 2219-2235

DOI

12
TitchmarshE C. The Theory of the Riemann Zeta-Function. 2nd ed. Oxford: Oxford University Press, 1986

13
WatsonG L. On indefinite quadratic forms in five variables. Proc Lond Math Soc, 1953, 3(3): 170-181

DOI

14
WooleyT D. On Weyl’s inequality, Hua’s lemma and exponential sums over binary forms. Duke Math J, 1999, 100: 373-423

DOI

15
WooleyT D. Vinogradov’s mean value theorem via efficient congruencing. Ann Math, 2012, 175: 1575-1627

DOI

Options
Outlines

/

〈 〉