2014/06/23 by Andrew R. Stout, Andrew Stout, Stout, Andrew
Chemistry · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Axial and Atropisomeric Chirality Synthesis #FOS: Mathematics #math.AG
paper · pdf · doi:10.48550/arxiv.1406.6083
40 pages, minor corrections, included explicit formula for auto-Igusa zeta function in the case of the cusp and the node (and nodal cubic)
openalex publication_date 2014/06/23 · arxiv created 2014/10/27 · arxiv updated 2014/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ∇_\mathfraknX provided that X is affine, \mathfrakn 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.