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- Title
Diophantine inequality involving binary forms.
- Authors
Mu, Quanwu
- Abstract
Let d ⩾ 3 be an integer, and set r = 2 + 1 for 3 ⩽ d ⩽ 4, $$\tfrac{{17}} {{32}} \cdot 2^d + 1$$ for 5 ⩽ d ⩽ 6, r = d + d+1 for 7 ⩽ d ⩽ 8, and r = d + d+2 for d ⩾ 9, respectively. Suppose that Φ( x, y) ∈ ℤ[ x, y] (1 ⩽ i ⩽ r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ, λ,..., λ are nonzero real numbers with λ/λ irrational, and λΦ( x , y ) + λΦ( x , y ) + · · · + λΦ( x , y ) is indefinite. Then for any given real η and σ with 0 < σ < 2, it is proved that the inequality has infinitely many solutions in integers x , x ,..., x , y , y ,..., y . This result constitutes an improvement upon that of B. Q. Xue.
- Subjects
DIOPHANTINE equations; MATHEMATICAL inequalities; NON-degenerate perturbation theory; INTEGERS
- Publication
Frontiers of Mathematics in China, 2017, Vol 12, Issue 6, p1457
- ISSN
1673-3452
- Publication type
Article
- DOI
10.1007/s11464-017-0602-y