Frontiers of Mathematics in China >
Diophantine inequality involving binary forms
Received date: 26 Jul 2016
Accepted date: 24 Aug 2016
Published date: 27 Nov 2017
Copyright
Let be an integer, and set and respectively. Suppose that are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ1, λ2, . . . , λr are nonzero real numbers with λ1/λ2 irrational, and λ1Φ1 (x1, y1) + λ2Φ2 (x2, y2) + · · · + λrΦr (xr, yr) is indefinite. Then for any given real η and σ with 0<σ<22−d, it is proved that the inequality has infinitely many solutions in integers x1, x2, . . . , xr, y1, y2, . . . , yr. This result constitutes an improvement upon that of B. Q. Xue.
Key words: Diophantine inequality; Davenport–Heilbronn method; binary form
Quanwu MU . Diophantine inequality involving binary forms[J]. Frontiers of Mathematics in China, 2017 , 12(6) : 1457 -1468 . DOI: 10.1007/s11464-017-0602-y
1 |
BourgainJ, DemeterC, GuthL. Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. Ann of Math, 2016, 184(2): 633–682
|
2 |
BrowningT D, DietmannR, P D T AElliott. Least zero of a cubic form. Math Ann, 2012, 352: 745–778
|
3 |
CookR J. The value of additive forms at prime arguments. J Théor Nombres Bordeaux, 2001, 13: 77–91
|
4 |
DavenportH, HeilbronnH. On indefinite quadratic forms in five variables. J Lond Math Soc, 1946, 21: 185–193
|
5 |
TitchmarshE C. The Theory of the Riemann Zeta-Function. 2nd ed. Oxford: Oxford Univ Press, 1986
|
6 |
VaughanR C. The Hardy-Littlewood method. Cambridge: Cambridge Univ Press, 1981
|
7 |
WatsonG L. On indefinite quadratic forms in five variables. Proc Lond Math Soc, 1953, 3(3): 170–181
|
8 |
Wooley T D. On Weyl’s inequality, Hua’s lemma, and exponential sums over binary forms. Duke Math J, 1999, 100: 373–423
|
9 |
WooleyT D. Hua’s lemma and exponential sums over binary forms. In: Rational Points on Algebraic Varieties. Basel: Birkhäuser, 2001, 405–446
|
10 |
WooleyT D. The cubic case of the main conjecture in Vinogradov’s mean value theorem. Adv Math, 2016, 294: 532–561
|
11 |
XueB Q. Diophantine inequality involving binary forms. Front Math China, 2014, 9(3): 641–657
|
/
〈 | 〉 |