Algebraic approximations in analytic geometry
Algebraic approximations in analytic geometry
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解析几何中的代数近似
DOI：
10.1007/bf01884302
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发表时间：
1995
影响因子：
3.1
通讯作者：
L. Lempert
中科院分区：
文献类型：
--
作者：
L. Lempert
设给定两个仿射代数簇V，W（始终在C上），一个紧集KCV，和一个全纯映射f：K~ W。(For本导言中的概念定义见第二节。2，8;也[GR 2，GR 3，Gu]）。我们问f是否可以一致近似的”代数”映射g：K-+ W。从龙格定理及其推广到多个变量（[H6]）的线索，很自然地假设K是全纯凸的（在V中）。然而，答案取决于我们所指的”代数”映射。一般来说，g不能被选择为多项式，例如当f（t）=（et，e-t）将平面中的单位圆盘映射为W={（Z1，Z2）：Z1 Z2 = 1} C C2时。在这种情况下，合理的近似是可能的，但在一般情况下，正确的概念”代数”功能和地图是一个纳什在[NS]。这些映射的分量满足多项式方程。我们的第一个结果是
Suppose we are given two affine algebraic varieties V, W (always over C), a compact set KCV, and a holomorphic mapping f: K~ W.(For definitions of notions in this introduction see Sects. 2, 8; also [GR2, GR3, Gu]). We ask whether f can be uniformly approximated by" algebraic" mappings g: K---+ W. Taking cue from Runge's theorem and its generalization to several variables ([H6]), it is natural to assume K is holomorphically convex (in V). The answer, however, depends on what we mean by" algebraic" mappings. In general g cannot be chosen as polynomial, eg when f (t)=(et, e-t) maps the unit disc in the plane into W={(Zl, Z2): ZlZ 2= 1} C C2. In this case rational approximation is possible, but in general the correct notion of" algebraic" functions and maps is the one given by Nash in [Ns]. These are mappings whose components satisfy polynomial equations. Our first result is