Thue inequalities with a small number of primitive solutions
Thue inequalities with a small number of primitive solutions
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具有少量原始解的不等式
DOI：
10.1023/a:1015217228291
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发表时间：
2001
影响因子：
0.8
通讯作者：
Kálmán Györi
中科院分区：
文献类型：
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作者：
Kálmán Györi
已知Thue方程（1）jF（x; y）j = m和更一般的Thue不等式（3）0 <jF（x; y）j m的本原解（x; y）的个数的几个上界。导出这样的上限的通常方法是区分“小”和“大”解，根据max（j x j ; j y j）小于或大于取决于F和m的适当显式常数Y;参见例如[1]，[11]，[6]和[2]。作为某些早期结果的改进和推广，我们在第1节中给出了（3）的本原解（x; y）个数的形式cn的上界，其中max（jxj; jyj）Y_0，其中c_（25）是常数，n表示所涉及的二元形式F的次数（参见定理1）。对于应用来说，重要的是，我们的大解的下界Y 0比[1]，[11]，[6]和[4]中的下界小得多，并且已经接近m的最佳可能性。利用定理1，我们在第2节中建立了（3）的原始解总数的类似上界，只要F的高度或判别式关于m足够大（参见图1）。定理2及其推论）。这些结果以定量的形式断言，在某种意义上，几乎所有形式（3）的不等式都只有很少的原始解。定理2及其结果是文[3]、[6]、[13]和[4]在这方面所得结果的相当大的改进。定理1和定理2的证明在第3节中给出。在证明中，我们对[1]和[6]的一些论点进行了适当的修改和补充。
Several upper bounds are known for the numbers of primitive solutions (x; y) of the Thue equation (1) j F(x; y) j = m and the more general Thue inequality (3) 0 < j F(x; y) j m. A usual way to derive such an upper bound is to make a distinction between "small" and "large" solutions, according as max( j x j ; j y j ) is smaller or larger than an appropriate explicit constant Y depending on F and m; see e.g. [1], [11], [6] and [2]. As an improvement and generalization of some earlier results we give in Section 1 an upper bound of the form cn for the number of primitive solutions (x; y) of (3) with max( j x j ; j y j )Y0 , wherec 25 is a constant and n denotes the degree of the binary form F involved (cf. Theorem 1). It is important for applications that our lower bound Y0 for the large solutions is much smaller than those in [1], [11], [6] and [4], and is already close to the best possible in terms of m. ByusingTheorem1 we establish in Section 2 similar upper bounds for the total number of primitive solutions of (3), provided that the height or discriminant of F is suficiently large with respect to m (cf. Theorem 2 and its corollaries). These results assert in a quantitative form that, in a certain sense, almost all inequalities of the form (3) have only few primitive solutions. Theorem 2 and its consequences are considerable improvements of the results obtained in this direction in [3], [6], [13] and [4]. The proofs of Theorems 1 and 2 are given in Section 3. In the proofs we use among other things appropriate modifications and refenements of some arguments of [1] and [6].