On a general Thue's equation
On a general Thue's equation
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DOI：
10.1353/ajm.2004.0034
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发表时间：
2004-09
影响因子：
1.7
通讯作者：
P. Corvaja;U. Zannier
中科院分区：
文献类型：
--
作者：
P. Corvaja;U. Zannier
本文分析了一类方程f 1···f r = g所定义的变量上的积分点，其中fi， g是n个变量的多项式，具有代数系数，且g具有“小”次；我们将使用最近在西格尔定理中介绍过的一种方法来求曲线上的积分点。经典的，非常特殊的，我们的方程的例子出现，例如，在一个著名的罗斯定理的推论（情况n = 2，如果i线性形式，deg r -2）和标准形式方程，由W. M.施密特处理。在这里，我们将证明（式1）积分点不是zariski稠密的，只要Σdeg fi n·max（度fi） +度g，并且假设fi， g满足某些“一般”验证的（温和）假设。我们的结论也涵盖了我们的超曲面的某些完全交子变种（表2）。最后，我们将证明（第3条）关于任意多项式代替线性形式的施密特子空间定理的一个类比。
We analyze the integral points on varieties defined by one equation of the form f 1 · · · f r = g , where the f i , g are polynomials in n variables with algebraic coefficients, and g has "small" degree; we shall use a method that we recently introduced in the context of Siegel's Theorem for integral points on curves. Classical, very particular, instances of our equations arise, e.g., in a well-known corollary of Roth's Theorem (the case n = 2, f i linear forms, deg g r -2) and with the norm-form equations , treated by W. M. Schmidt. Here we shall prove (Thm. 1) that the integral points are not Zariski-dense, provided Σdeg f i n · max (deg f i ) + deg g and provided the f i , g , satisfy certain (mild) assumptions which are "generically" verified. Our conclusions also cover certain complete-intersection subvarieties of our hypersurface (Thm. 2). Finally, we shall prove (Thm. 3) an analogue of the Schmidt's Subspace Theorem for arbitrary polynomials in place of linear forms.