混合幂丢番图不等式（英文）

It is proved that if λ_1, λ_2, ···, λ_7are nonzero real numbers, not all of the same sign and not all in rational ratios, then for any given real numbers η and σ, 0 σ 1/16, the inequality |λ_1x~2_1+ λ_2x~2_2+∑7 i=3λ_ix~4_i+ η| ( max1≤i≤7|x_i|)~(-σ)has infinitely many solutions in positive integers x_1, x_2, ···, x_7. Similar result is proved for |λ_1x~2_1+ λ_2x~2_2+ λ_3x~2_3+ λ_4x~4_4+ λ_5x~4_5+ λ_6x~4_6+ η| ( max1≤i≤6|xi|)-σ.These results constitute an improvement upon those of Shi and Li.