Diophantine inequality involving binary forms

Boqing XUE

Front. Math. China ›› 2014, Vol. 9 ›› Issue (3) : 641-657.

PDF(165 KB)
PDF(165 KB)
Front. Math. China ›› 2014, Vol. 9 ›› Issue (3) : 641-657. DOI: 10.1007/s11464-013-0334-6
RESEARCH ARTICLE
RESEARCH ARTICLE

Diophantine inequality involving binary forms

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Abstract

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

|i=1rλiΦi(xi,yi)+η||<(max1ir{|xi|,|yi|})-σ
where Φi(x, y) ∈Z[x, y] (1≤ir) are nondegenerate forms of degree d= 3 or 4.

Keywords

Diophantine inequality / binary form

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Boqing XUE. Diophantine inequality involving binary forms. Front. Math. China, 2014, 9(3): 641‒657 https://doi.org/10.1007/s11464-013-0334-6

References

[1]
BakerR C. Cubic diophantine inequalities. Mathematika, 1982, 29: 83-92
CrossRef Google scholar
[2]
BakerR C, BrüdernJ, WooleyT D. Cubic diophantine inequalities. Mathematika, 1995, 42: 264-277
CrossRef Google scholar
[3]
BrowningT D. Quantitative Arithmetic of Projective Varieties. Progress in Math, Vol 277. Basel: Birkhäuser, 2009
CrossRef Google scholar
[4]
BrowningT D, DietmannR, ElliottP D T A. Least zero of a cubic form. Math Ann, 2012, 352: 745-778
CrossRef Google scholar
[5]
BrüdernJ. Cubic diophantine inequalities. Mathematika, 1988, 35: 51-58
CrossRef Google scholar
[6]
BrüdernJ. Cubic diophantine inequalities (II). J Lond Math Soc, 1996, 53(2): 1-18
CrossRef Google scholar
[7]
BrüdernJ. Cubic diophantine inequalities (III). Periodica Mathematica Hungarica, 2001, 42(1-2): 211-226
CrossRef Google scholar
[8]
CookR J. The value of additive forms at prime arguments. J Théor Nombres Bordeaux, 2001, 13: 77-91
CrossRef Google scholar
[9]
DavenportH. Analytic Methods for Diophantine Equations and Diophantine Inequalities. 2nd ed. Cambridge: Cambridge University Press, 2005
CrossRef Google scholar
[10]
DavenportH, HeilbronnH. On indefinite quadratic forms in five variables. J Lond Math Soc, 1946, 21: 185-193
CrossRef Google scholar
[11]
HarveyM P. Cubic diophantine inequalities involving a norm form. Int J Number Theory, 2011, 7(8): 2219-2235
CrossRef Google scholar
[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
CrossRef Google scholar
[14]
WooleyT D. On Weyl’s inequality, Hua’s lemma and exponential sums over binary forms. Duke Math J, 1999, 100: 373-423
CrossRef Google scholar
[15]
WooleyT D. Vinogradov’s mean value theorem via efficient congruencing. Ann Math, 2012, 175: 1575-1627
CrossRef Google scholar

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