On the auto Igusa-zeta function of an algebraic curve
On the auto Igusa-zeta function of an algebraic curve
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关于代数曲线的 auto Igusa-zeta 函数
DOI：
10.1016/j.jsc.2016.08.011
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Andrew R. Stout
中科院分区：
文献类型：
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作者：
Andrew R. Stout
我们研究了完备Noether局部环在motivic积分下的自同态。使用自弧空间的概念，我们在一个点上引入（约化的）自Igusa zeta级数，它似乎可以衡量一个品种在该点上不光滑的程度。我们猜想一个封闭的公式的情况下，曲线的一个奇点，我们提供了明确的公式，这一系列的情况下的尖点和节点。使用Denef和Loeser的工作，可以证明这个系列通常是理性的。这些想法是通过Sage中的大量计算获得的。因此，我们包括在这些计算中使用的Sage脚本。它计算仿射弧空间<$n X，条件是X是仿射的，n是胖点，并且基场是特征零。最后，我们表明，自动庞加莱系列往往是合理的，以及连接到有关新类型的motivic积分的问题。
We study endomorphisms of complete Noetherian local rings in the context of motivic integration. Using the notion of an auto-arc space, we introduce the (reduced) auto-Igusa zeta series at a point, which appears to measure the degree to which a variety is not smooth that point. We conjecture a closed formula in the case of curves with one singular point, and we provide explicit formulas for this series in the case of the cusp and the node. Using the work of Denef and Loeser, one can show that this series will often be rational. These ideas were obtained through extensive calculations in Sage. Thus, we include a Sage script which was used in these calculations. It computes the affine arc spaces∇ n X provided that X is affine, n is a fat point, and the ground field is of characteristic zero. Finally, we show that the auto Poincaré series will often be rational as well and connect this to questions concerning new types of motivic integrals.