A rigid analytic version of M. Artin's theorem on analytic equations
A rigid analytic version of M. Artin's theorem on analytic equations
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M. Artin 解析方程定理的严格解析版本
DOI：
10.1007/bf01450712
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发表时间：
1981
期刊：
影响因子：
--
通讯作者：
Siegfried Bosch
中科院分区：
文献类型：
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作者：
Siegfried Bosch
专业化定理。令 {Pi} i~ 为 R [Y] 中的多项式系统，其中 Y=(Y1,..., I1,) 表示一组不定式。假设存在一个坐标为R的点Y=(Yi,..-,Y,)，解方程pi(Y)=0，即I。然后可以找到任意接近于在 R 中具有坐标的 y 的解 y=(Yl..... y,)。取 n= 1 并考虑一对积分域，只有当 R 是代数闭于 /~ 时，特殊化定理才成立，即，如果 /~-R 中没有元素是 R [Y1] 中非平凡多项式的零。 Lang 推测这个条件对于“自然”对 RC/~ 也是充分的。特殊化定理在以下情况下成立：(i)/~是特征零域，带有完全估值，R是稠密子域，代数上闭于/~（参见Lang[6，定理11]）。
Specialization Theorem. Let {Pi} i~ be a system of polynomials in R [Y] where Y=(Y1,..., I1,) denotes a set of indeterminates. Assume that there exists a point Y=(Yi,..-, Y,) with coordinates in R solving the equations pi (Y)= 0, ieI. Then one can find solutions y=(Yl..... y,) arbitrarily close to y having coordinates in R.Taking n= 1 and considering a pair of integral domains, the Specialization Theorem can only be true if R is algebraically closed in/~, ie, if no element in/~-R is a zero of a non-trivial polynomial in R [Y1]. That this condition is also sufficient was conjectured by Lang for" natural" pairs RC/~. The Specialization Theorem has been established in the following cases:(i)/~ is a field of characteristic zero, carrying a complete valuation, and R is a dense subfield, algebraically closed in/~(see Lang [6, Theorem 11]).