Applications of Diophantine Approximation to Integral Points and Transcendence
Applications of Diophantine Approximation to Integral Points and Transcendence
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丢番图逼近在积分点和超越中的应用
DOI：
10.1017/9781108348096
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发表时间：
2018
影响因子：
1.9
通讯作者：
U. Zannier
中科院分区：
文献类型：
--
作者：
P. Corvaja;U. Zannier
丢番图近似可以粗略地描述为数论中与有理数近似有关的分支;或者更确切地说，这构成了最初的动机。这些问题吸引了持续的关注，无疑是由于他们的相关性，另一个更古老的话题：理论的丢番图方程，即那些解决方案必须找到整数或有理数，可能在有限的扩展Q。这些主题之间的联系，已经被拉格朗日和勒让德观察到，被挪威的A。图厄; 1909年，他证明了有限定理丢番图方程，其中第一次包括整个家庭的方程，任意大的程度。当时，他们只能偶尔得到治疗，而且只能用特别的方法，尽管是巧妙的方法。图厄定理仅仅依赖于一个结果，限制了准确性的合理近似代数数（以前的结果已获得刘维，但它太弱的应用方程）。
Diophantine approximation may be roughly described as the branch of number theory concerned with approximations by rational numbers; or rather, this constituted the original motivation. That such questions have attracted continued attention is undoubtedly substantially due to their relevance for another, more ancient, topic: the theory of Diophantine equations, namely those whose solutions have to be found in integers or rationals, possibly in a finite extension of Q. The connections between the subjects, which had already been observed by Lagrange and Legendre, were explicitly pointed out by the Norwegian A. Thue; in 1909 he proved a finiteness theorem for Diophantine equations which for the first time included whole families of equations, of arbitrarily large degree. At that time they could be treated only occasionally, and merely with ad hoc methods, albeit ingenious ones. Thue’s theorem relied solely on a result which limited the accuracy of the rational approximations to algebraic numbers (a previous result had been obtained by Liouville, but it was too weak for applications to equations).