Diophantine inequality involving binary forms

doi:10.1007/s11464-017-0602-y

Front. Math. China

2017, Vol. 12

Issue (6) : 1457-1468 https://doi.org/10.1007/s11464-017-0602-y

RESEARCH ARTICLE

Diophantine inequality involving binary forms

Quanwu MU(

)

School of Science, Xi’an Polytechnic University, Xi’an 710048, China

Download: PDF(163 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

Let $d ≥ 3$ be an integer, and set $r = 2 d − 1 + 1 ? for ? 3 ≤ d ≤ 4,$ $r = 1732 ? 2 d + 1 ? for ? 5 ≤ d ≤ 6,$ $r = d 2 + d + 1 ? for ? 7 ≤ d ≤ 8,$ and $r = d 2 + d + 2 ? for d ≥ 9,$ respectively. Suppose that $Φ i (x, y) ∈ ? | x, y | (1 ≤ i ≤ r)$ are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ₁, λ₂, . . . , λ_r are nonzero real numbers with λ₁/λ₂ irrational, and λ₁Φ₁ (x₁, y₁) + λ₂Φ₂ (x₂, y₂) + · · · + λ_rΦ_r (x_r, y_r) 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, . . . , x_r, y₁, y₂, . . . , y_r. This result constitutes an improvement upon that of B. Q. Xue.

Keywords Diophantine inequality Davenport–Heilbronn method binary form

Corresponding Author(s): Quanwu MU

Issue Date: 27 November 2017

Cite this article:

Quanwu MU. Diophantine inequality involving binary forms[J]. Front. Math. China, 2017, 12(6): 1457-1468.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-017-0602-y
https://academic.hep.com.cn/fmc/EN/Y2017/V12/I6/1457

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 https://doi.org/10.4007/annals.2016.184.2.7
2	BrowningT D, DietmannR, P D T AElliott. Least zero of a cubic form. Math Ann, 2012, 352: 745–778 https://doi.org/10.1007/s00208-011-0651-6
3	CookR J. The value of additive forms at prime arguments. J Théor Nombres Bordeaux, 2001, 13: 77–91 https://doi.org/10.5802/jtnb.305
4	DavenportH, HeilbronnH. On indefinite quadratic forms in five variables. J Lond Math Soc, 1946, 21: 185–193 https://doi.org/10.1112/jlms/s1-21.3.185
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 https://doi.org/10.1112/plms/s3-3.1.170
8	Wooley T D. On Weyl’s inequality, Hua’s lemma, and exponential sums over binary forms. Duke Math J, 1999, 100: 373–423 https://doi.org/10.1215/S0012-7094-99-10014-7
9	WooleyT D. Hua’s lemma and exponential sums over binary forms. In: Rational Points on Algebraic Varieties. Basel: Birkhäuser, 2001, 405–446 https://doi.org/10.1007/978-3-0348-8368-9_15
10	WooleyT D. The cubic case of the main conjecture in Vinogradov’s mean value theorem. Adv Math, 2016, 294: 532–561 https://doi.org/10.1016/j.aim.2016.02.033
11	XueB Q. Diophantine inequality involving binary forms. Front Math China, 2014, 9(3): 641–657 https://doi.org/10.1007/s11464-013-0334-6

[1]	Gaiyun GAO, Zhixin LIU. Results of Diophantine approximation by unlike powers of primes[J]. Front. Math. China, 2018, 13(4): 797-808.
[2]	Yuchao WANG. Values of binary linear forms at prime arguments[J]. Front. Math. China, 2015, 10(6): 1449-1459.
[3]	Boqing XUE. Diophantine inequality involving binary forms[J]. Front. Math. China, 2014, 9(3): 641-657.

Viewed

Full text

Abstract

Cited

Shared

Discussed