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On real log canonical thresholds

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arxiv 0707.2308 v3 pith:Q7OJBINT submitted 2007-07-16 math.AG

Morihiko Saito This is my paper · ORCID
classification math.AG
keywords realjumpingcanonicalfunctionnumbernumberssignthreshold
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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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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Archimedean zeta functions, singularities, and Hodge theory

    math.AG 2024-12 accept novelty 8.0 of 10

    The largest nontrivial pole of the Archimedean zeta function equals the negative of the minimal exponent, with exact multiplicity, while some other Bernstein-Sato roots are not poles.

  2. Divisorial Persistence and Asymptotic Homology of Analytic Pairs

    math.AG 2026-07 reject novelty 7.0 of 10

    Divisorial asymptotic homology is defined so that its minimum and jump locus recover the RLCT, but the stated filtration also admits chains disjoint from the singular locus, invalidating the claimed zero-below-thresho...

  3. A lower bound on the analytic log-canonical threshold over local fields of positive characteristic

    math.AG 2025-11 conditional novelty 6.0 of 10

    For local fields of positive characteristic, every nonzero analytic function has strictly positive integrability threshold, and for regular functions on smooth varieties the threshold is at least 1/(d·D^m).

  4. Subtlety of oscillation indices of oscillatory integrals of real analytic functions

    math.CV 2025-11 reject novelty 4.0 of 10

    For real analytic homogeneous functions that are Newton R-nondegenerate and convenient, the paper claims a strict inequality (even n) or equality (odd n) between the absolute oscillation index and the real log canonic...

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