IGUSA’S CONJECTURE FOR EXPONENTIAL SUMS: OPTIMAL ESTIMATES FOR NONRATIONAL SINGULARITIES

Forum of Mathematics Pi ◽

10.1017/fmp.2019.3 ◽

2019 ◽

Vol 7 ◽

Cited By ~ 2

Author(s):

RAF CLUCKERS ◽

MIRCEA MUSTAŢĂ ◽

KIEN HUU NGUYEN

Keyword(s):

Upper Bound ◽

Exponential Sums ◽

Log Canonical Threshold ◽

Log Canonical ◽

Optimal Estimates

We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa’s conjecture on exponential sums, with the log canonical threshold in the exponent of the estimates. We show that this covers optimally all situations of the conjectures for nonrational singularities by comparing the log canonical threshold with a local notion of the motivic oscillation index.

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Tighter upper bound of real log canonical threshold of non-negative matrix factorization and its application to Bayesian inference

2017 IEEE Symposium Series on Computational Intelligence (SSCI) ◽

10.1109/ssci.2017.8280811 ◽

2017 ◽

Cited By ~ 1

Author(s):

Naoki Hayashi ◽

Sumio Watanabe

Keyword(s):

Bayesian Inference ◽

Upper Bound ◽

Matrix Factorization ◽

Log Canonical Threshold ◽

Log Canonical ◽

Non Negative Matrix Factorization

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Proof of the Cluckers–Veys Conjecture on Exponential Sums for Polynomials With Log-Canonical Threshold at Most a Half

International Mathematics Research Notices ◽

10.1093/imrn/rnz036 ◽

2019 ◽

Cited By ~ 1

Author(s):

Saskia Chambille ◽

Kien Huu Nguyen

Keyword(s):

Exponential Sums ◽

Log Canonical Threshold ◽

Log Canonical

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Floer cohomology, multiplicity and the log canonical threshold

Geometry & Topology ◽

10.2140/gt.2019.23.957 ◽

2019 ◽

Vol 23 (2) ◽

pp. 957-1056

Author(s):

Mark McLean

Keyword(s):

Floer Cohomology ◽

Log Canonical Threshold ◽

Log Canonical

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Learning Coefficient of Vandermonde Matrix-Type Singularities in Model Selection

Entropy ◽

10.3390/e21060561 ◽

2019 ◽

Vol 21 (6) ◽

pp. 561

Author(s):

Miki Aoyagi

Keyword(s):

Model Selection ◽

Information Criteria ◽

Learning Systems ◽

Vandermonde Matrix ◽

Selection Methods ◽

Learning Models ◽

Matrix Type ◽

Log Canonical Threshold ◽

Log Canonical ◽

Blowing Up

In recent years, selecting appropriate learning models has become more important with the increased need to analyze learning systems, and many model selection methods have been developed. The learning coefficient in Bayesian estimation, which serves to measure the learning efficiency in singular learning models, has an important role in several information criteria. The learning coefficient in regular models is known as the dimension of the parameter space over two, while that in singular models is smaller and varies in learning models. The learning coefficient is known mathematically as the log canonical threshold. In this paper, we provide a new rational blowing-up method for obtaining these coefficients. In the application to Vandermonde matrix-type singularities, we show the efficiency of such methods.

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Canonical and log canonical thresholds of multiple projective spaces

European Journal of Mathematics ◽

10.1007/s40879-019-00388-7 ◽

2019 ◽

Author(s):

Aleksandr V. Pukhlikov

Keyword(s):

Projective Space ◽

Projective Spaces ◽

Regularity Conditions ◽

Large Classes ◽

Log Canonical Threshold ◽

Birational Rigidity ◽

Log Canonical ◽

Finite Set ◽

Dimensional Projective Space ◽

Almost All

AbstractWe show that the global (log) canonical threshold of d-sheeted covers of the M-dimensional projective space of index 1, where $$d\geqslant 4$$d⩾4, is equal to 1 for almost all families (except for a finite set). The varieties are assumed to have at most quadratic singularities, the rank of which is bounded from below, and to satisfy the regularity conditions. This implies birational rigidity of new large classes of Fano–Mori fibre spaces over a base, the dimension of which is bounded from above by a constant that depends (quadratically) on the dimension of the fibre only.

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