On the number of solutions of polynomial congruences and Thue equations
On the number of solutions of polynomial congruences and Thue equations
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DOI：
10.1090/s0894-0347-1991-1119199-x
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发表时间：
1991-01
影响因子：
3.9
通讯作者：
C. Stewart
中科院分区：
文献类型：
--
作者：
C. Stewart
只有有限个整数x和yv的解。本文第一部分在F的判别式D(F)不为零的假设下，建立了(1)在素数整数x和y中解的个数的上界。对于大多数整数h，这些界是Bombieri和Schmidt在[5]中得到的界的改进。在证明这些边界的过程中，我们将建立一个关于多项式同余的结果，该结果扩展了Nagell [30]， Ore [32]， Sandor[33]和Huxley[19]的早期工作。事实上，我们要建立一个多项式同余解的个数的上界，一般来说，它是最好的可能。在第二部分中，我们将解决寻找形式F的问题，其中(1)对任意大整数h有许多解。最后，我们将通过利用Evertse， Gy6ry， Stewart和Tijdeman[17]对s -单位方程解的数量的估计，获得某些Thue-Mahler和Ramanujan-Nagell方程解的数量的上界。
has only finitely many solutions in integers x and yv. In the first part of this paper we shall establish upper bounds for the number of solutions of (1) in coprime integers x and y under the assumption that the discriminant D(F) of F is nonzero. For most integers h these bounds improve upon those obtained by Bombieri and Schmidt in [5]. In the course of proving these bounds we shall establish a result on polynomial congruences that extends earlier work of Nagell [30], Ore [32], Sandor [33], and Huxley [19]. In fact we shall establish an upper bound for the number of solutions of a polynomial congruence that is, in general, best possible. In the second part we shall address the problem of finding forms F for which (1) has many solutions for arbitrarily large integers h. Finally we shall obtain upper bounds for the number of solutions of certain Thue-Mahler and Ramanujan-Nagell equations by appealing to estimates of Evertse, Gy6ry, Stewart, and Tijdeman [17] for the number of solutions of S-unit equations.