On real log canonical thresholds

2007/07/16 by Morihiko Saito, Saito, Morihiko · 4 citations
Computer Science · Mathematics · #32S40 #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.0707.2308

openalex publication_date 2007/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce real log canonical threshold and real jumping numbers for real algebraic functions. A real jumping number is a root of the b-function up to a sign if its difference with the minimal one is less than 1. The real log canonical threshold, which is the minimal real jumping number, coincides up to a sign with the maximal pole of the distribution defined by the complex power of the absolute value of the function. However, this number may be greater than 1 if the codimension of the real zero locus of the function is greater than 1. So it does not necessarily coincide with the maximal root of the b-function up to a sign, nor with the log canonical threshold of the complexification. In fact, the real jumping numbers can be even disjoint from the non-integral jumping numbers of the complexification.

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• A lower bound on the analytic log-canonical threshold over local fields of positive characteristic
• Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory
• Desingularization in Computational Applications and Experiments
• An asymptotic approximation of the marginal likelihood for general\n Markov models

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