Diophantine inequality involving binary forms |
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Authors: | Quanwu Mu |
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Affiliation: | 1.School of Science,Xi’an Polytechnic University,Xi’an,China |
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Abstract: | Let d ? 3 be an integer, and set r = 2 d?1 + 1 for 3 ? d ? 4, (tfrac{{17}}{{32}} cdot 2^d + 1) for 5 ? d ? 6, r = d2+ d+1 for 7 ? d ? 8, and r = d2+ 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 λ 1, λ 2,..., λ r are nonzero real numbers with λ 1/λ 2 irrational, and λ 1Φ 1( x1, y1) + λ 2Φ 2( x2, y2) + · · · + λ r Φ r ( x r , y r ) is indefinite. Then for any given real η and σ with 0 < σ < 2 2?d, it is proved that the inequality $$left| {sumlimits_{i = 1}^r {{lambda _i}Phi {}_ileft( {{x_i},{y_i}} right) + eta } } right| < {left( {mathop {max left{ {left| {{x_i}} right|,left| {{y_i}} right|} right}}limits_{1 leqslant i leqslant r} } right)^{ - sigma }}$$ 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. |
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