Integration of semialgebraic functions and integrated Nash functions
Integration of semialgebraic functions and integrated Nash functions
复制标题
半代数函数和积分纳什函数的积分
DOI：
10.1007/s00209-012-1138-1
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发表时间：
2013
影响因子：
0.8
通讯作者：
T. Kaiser
中科院分区：
文献类型：
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作者：
T. Kaiser
当积分半代数函数时，必须离开半代数设置。例如，可以得到诸如反正切函数等代数幂函数级数的全局对数和迭代反导数。我们证明了用这些函数来扩展半代数函数就足以完全描述半代数函数的参数积分。为了实现这一点，我们在取反导数的情况下闭合了任意维代数幂级数的环。我们对这些环进行了深刻的分析。特别地，我们证明了韦尔斯特拉斯除法定理和韦尔斯特拉斯预备定理成立。这使得我们可以应用模型理论的结果来获得半代数函数的参数积分的显式描述。最后，我们研究了由积分代数幂级数生成的结构。
When integrating semialgebraic functions one has to leave the semialgebraic setting. For example, one gets the global logarithm and iterated antiderivatives of algebraic power series such as the arctangent. We show that it is enough to enlarge the semialgebraic functions by these functions to completely describe parameterized integrals of semialgebraic functions. To realize this we close the rings of algebraic power series in arbitrary dimension under taking antiderivatives. We analyze these rings profoundly. In particular we show that the Weierstrass division theorem and the Weierstrass preparation theorem hold. This allows us to apply model theoretic results to obtain an explicit description of parameterized integrals of semialgebraic functions. Finally, we investigate the structure generated by the integrated algebraic power series.