Infinite differentiability in polynomially bounded o-minimal structures
Infinite differentiability in polynomially bounded o-minimal structures
复制标题
多项式有界 o 极小结构中的无限可微性
DOI：
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发表时间：
1995
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通讯作者：
Chris Miller
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文献类型：
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作者：
Chris Miller
定义在有序真实的数域的多项式有界o-极小展开式中的不可微函数具有真实的解析函数的一些优良性质。特别地，如果一个可定义函数f：Rn R在a ∈ Rn对所有N ∈ N是CN，并且f的所有偏导数在a处为零，则f在a的某个开邻域上相同地为零。结合这与Abhyankar-Moh定理收敛的幂级数，它表明，如果9 t是一个多项式有界的o-最小的扩展领域的真实的号码与限制解析功能，那么所有的Cc* 功能定义在91是真实的解析，提供这是真的所有可定义的功能的一个变量。
Infinitely differentiable functions definable in a polynomially bounded o-minimal expansion 9l of the ordered field of real numbers are shown to have some of the nice properties of real analytic functions. In particular, if a definable function f: Rn R is CN at a E Rn for all N E N and all partial derivatives of f vanish at a, then f vanishes identically on some open neighborhood of a . Combining this with the Abhyankar-Moh theorem on convergence of power series, it is shown that if 9t is a polynomially bounded o-minimal expansion of the field of real numbers with restricted analytic functions, then all Cc* functions definable in 91 are real analytic, provided that this is true for all definable functions of one variable.