一个混合幂的丢番图不等式(英文)

On a Diophantine Inequality With Mixed Powers

本文证明了如果λ_1,λ_2,…,λ_8为不全同号的非零实数,其中λ_1/λ_2为无理数,则对任意实数κ及0<σ<1/8,不等式|λ_1x_1~2+λ_2x_2~2+sum ( λ_ix_i^4+κ) from i=3 to 8|<(max 1≤i≤8 xi)^(-σ)有无穷多组整数解(x_1,x_2,…,x_8).

In this paper, it is proved that if λ1, λ2,…, λ8 are nonzero real numbers, not all of the same sign, and λ1/λ2 is irrational, then for any real number κ and 0〈σ〈1/8, the inequality ｜λ1x1^2＋λ2x2^2＋∑（i=3→8）λixi^4＋κ｜〈（max（1≤i≤8）x1）^-σ has infinitely many solutions in positive integers x1, x2,… ,x8. This result constitutes an improvement upon that of Gong and Li.