Exponential rings, exponential polynomials and exponential functions
Exponential rings, exponential polynomials and exponential functions
复制标题
指数环、指数多项式和指数函数
DOI：
10.2140/pjm.1984.113.51
复制
发表时间：
1984
影响因子：
0.6
通讯作者：
L. Dries
中科院分区：
文献类型：
--
作者：
L. Dries
导论.一个指数环，或简称为E-环，是一个对（i？，E），其中R是环-在本文中总是与1-交换的，E是R的加法群到R的单位乘法群的态射，即对i？中的所有x9，y，E（x + y）= E（x）E（y），且E（0）= 1。例子是（R，a），a任何正的真实的，和（C，e）.当然，任何环R都可以通过对所有x\置E（x）-1而扩展为ε-环（R，E），这样的环将被称为平凡环。Ken Manders观察到一个Zi-环，其基础环没有幂零元φ 0，并且特征有一个素数f？> 0是平凡的：在这样的环中，每个x满足1 = E（0）-E（px）= E（x），所以（E（x）\y = 0，这意味着E（x）= 1。指数环的有关概念M.比森，由B。Dahn和Wolter，以及A.威尔基，所有这些都与A.塔斯基论实数域的幂可判定性。如果没有超越数论的重大进展，这个问题的有效正解似乎是不可能的：这样的解将给我们一个回答任何问题的决策方法：是e = p/q>>其中/？，q是正整数。当然有这样一种决策方法，但是，由于我们还不知道e是否是理性的，我们不知道它是如何工作的。现在在数学实践中，它是少的有效性塔斯基的决定方法的真实的领域的事项-虽然这方面是有趣的-而是信息的方法提供的代数拓扑性质的可定义集在R，并对渐近行为的可定义的职能。例如，在半代数和真实的代数几何中，这种用法在塔斯基-塞登伯格定理（非建设性版本）中得到了形式化，并在一个结果中得到了类似于半代数集的连通分量的数量有限性的结果。部分使用塔斯基的工作的基本理论的现实提供更多的希望被推广到E-ng（R，e）。以下
Introduction. An exponential ring, or E-ring for short, is a pair (i?, E) with R a ring—in this paper always commutative with 1—and E a morphism of the additive group of R into the multiplicative group of units of R, that is, E(x + y) = E(x)E(y) for all x9 y in i?, and E(0) = 1. Examples are (R, a), a any positive real, and (C, e). Of course, any ring R can be expanded to an £-ring (R, E) by putting E(x) — 1 for all x\ such brings will be called trivial. Ken Manders observed that an Zί-ring whose underlying ring has no nilpotents φ 0 and has characteristic a prime/? > 0 is trivial: in such a ring each x satisfies 1 = E(0) — E(px) = E{x), so (E(x) \y = 0, which implies E(x) = 1. Related notions of exponential ring have been considered by M. Beeson, by B. Dahn and Wolter, and by A. Wilkie, all in connection with the longstanding open problem of A. Tarski on the decidability of the field of reals with exponentiation. An effective positive solution to this problem seems unlikely without major advances in transcendental number theory: such a solution would give us a decision method to answer any question: is e = p/q>> where/?, q are positive integers. Of course there is such a decision method, but, as we don't know yet whether e is rational, we don't know how it works. Now in mathematical practice it is less the effectiveness of Tarski's decision method for the real field which matters—though this aspect is interesting—but rather the information the method provides on the algebraic-topological nature of the definable sets in R, and on the asymptotic behavior of definable functions. For example in semi-algebraic and real algebraic geometry this use is formalized in the Tarski-Seidenberg theorem (in an inconstructive version) and in a result like the finiteness of the number of connected components of a semi-algebraic set. Parts of this use of Tarski's work on the elementary theory of the reals offer more hope of being generalized to the E-ήng (R, e). The following