COUNTING SOLUTIONS OF ∣axr–byr∣≤h
COUNTING SOLUTIONS OF ∣axr–byr∣≤h
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∣axr–byr∣≤h 的计算解
DOI：
10.1093/qmath/38.4.503
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发表时间：
1987
影响因子：
0.7
通讯作者：
J. Mueller
中科院分区：
文献类型：
--
作者：
J. Mueller
F(x，y)=h，(1.1)其中F是具有整系数的r5=3次形式，在有理数上是不可约的，只有有限多个解。很久以后，Baker[1]给出了解的大小的明确的，但相当大的界。解的数目的界在前面已经给出了(例如Lewis和Mahler[7])，但Evertse[5]是第一个仅用r和h来估计解的数目的人。Bombieri和Schmidt最近给出了更好的这种界限[4]。这里我们将讨论二进制形式F=axr-byr的特殊情况。对于这类形式的一般估计可以大大减少，从而为Siegel的猜想提供了一些证据，即解的数目可以关于h和F中的单项式的数目是有界的。更准确地说，我们的结果不仅是关于AXR-BYR=h的解，而且是GCD{x，v)=1的丢番图\AXR-BYR\=GB h的解。
F (x, y)= h,(1.1) where F is a form of degree r 5= 3 with integral coefficients, and irreducible over the rationals, has only finitely many solutions. Much later Baker [1] gave explicit, but rather large, bounds for the size of the solutions. Bounds for the number of solutions had been given earlier (see eg Lewis and Mahler [7]), but Evertse [5] was the first to estimate the number of solutions in terms of r and h only. Much better such bounds were recently given by Bombieri and Schmidt [4]. Here we shall deal with the special case of a binary form F= axr-byr. The general estimates can be greatly reduced for forms of this type, giving some evidence to Siegel's conjecture that the number of solutions may be bounded in terms of h and the number of monomials in F. More precisely, our results will be on solutions not only of axr-byr= h, but rather of the diophantine inequality\axr-byr\=£ h with gcd {x, v)= 1.(1.2)