Characteristic one, entropy and the absolute point
Characteristic one, entropy and the absolute point
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特征一、熵与绝对点
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
C. Consani
中科院分区：
文献类型：
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作者：
A. Connes;C. Consani
我们表明，数学意义的工作特征之一是直接连接到领域的幂等分析和热带代数几何，我们将这个想法的绝对点的概念。在引入特征1的“完美”半环的概念后，说明了如何使正特征的Witt环的构造适应于特征1的极限情形。这个结构揭示了熵和热力学之间有趣的联系，同时也为经典的维特结构本身提供了新的线索。我们简化了F_1-格式的几何实现的构造，并将我们以前对zeta函数的计算推广到含挠的F_1-格式的情形。然后，我们表明，添加剂结构的研究提供了一个自然的映射，从monoids集接近满足要求的假设曲线紧规格Z的绝对点。最后，我们测试了有理数上椭圆曲线上zeta函数的计算。
We show that the mathematical meaning of working in characteristic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion of the absolute point. After introducing the notion of "perfect" semi-ring of characteristic one, we explain how to adapt the construction of the Witt ring in positive characteristic to the limit case of characteristic one. This construction unveils an interesting connection with entropy and thermodynamics, while shedding a new light on the classical Witt construction itself. We simplify our earlier construction of the geometric realization of an F_1-scheme and extend our earlier computations of the zeta function to cover the case of F_1-schemes with torsion. Then, we show that the study of the additive structures on monoids provides a natural map from monoids to sets which comes close to fulfill the requirements for the hypothetical curve compactifying Spec Z over the absolute point. Finally, we test the computation of the zeta function on elliptic curves over the rational numbers.