Diophantine inequality involving binary forms

Abstract

Let d\ge \mathrm{3} be an integer, and set r={\mathrm{2}}^{d-\mathrm{1}}+\mathrm{1}\hspace{0.17em}\mathrm{for}\hspace{0.17em}\mathrm{3}\le d\le \mathrm{4},r={\displaystyle \frac{\mathrm{17}}{\mathrm{32}}}\cdot {\mathrm{2}}^d+\mathrm{1}\hspace{0.17em}\mathrm{for}\hspace{0.17em}5\le d\le \mathrm{6},r=d^{\mathrm{2}}+d+\mathrm{1}\hspace{0.17em}\mathrm{for}\hspace{0.17em}\mathrm{7}\le d\le \mathrm{8}, and r=d^{\mathrm{2}}+d+\mathrm{2}\hspace{0.17em}\mathrm{for }d\ge \mathrm{9}, respectively. Suppose that {{\displaystyle \Phi }}_i(x,y)\in \mathbb{Z}\left|x,y\right|(\mathrm{1}\le i\le 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.

Key words： Diophantine inequality; Davenport–Heilbronn method; binary form

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