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Hiromasa Nakayama

Publications and source records attributed to Hiromasa Nakayama.

9 recordsLinked to original sources
arXiv2026

Comprehensive Restriction Algorithm for Hypergeometric Systems

An algorithm computing the restriction of a holonomic D-module to a linear subspace was given by T.Oaku in 1997. We consider a problem of computing the restriction for a given holonomic D-module with parameters. We will give a partial answer to the problem for general holonomic D-modules and an answer to hypergeometric holonomic D-modules. The above results are based on comprehensive Groebner system and an algorithm for the isomorphic classification of hypergeometric D-modules.

math.AC↗
arXiv2026

Flat coordinates of Frobenius prepotentials related with the reflection groups of types $H_3$ and $H_4$

In this article, we first explain a group theoretic interpretation of the derivation of the relation between the flat coordinates of the polynomial prepotential $(H_3)$ and those of the algebraic prepotential $(H_3)'$ given in \cite{KMS2} constructed by M. Feigin, D. Valeri and J. Wright \cite{FVW}. By the same idea explained in the case of $(H_3)$, we will show a relation between the flat coordinates of the polynomial prepotential $(H_4)$ and those of the algebraic prepotential $H_4(9)$ given in \cite{Se}.

math.AC↗
arXiv2013

Holonomic Gradient Descent for the Fisher-Bingham Distribution on the $d$-dimensional Sphere

We propose an accelerated version of the holonomic gradient descent and apply it to calculating the maximum likelihood estimate (MLE) of the Fisher-Bingham distribution on a $d$-dimensional sphere. We derive a Pfaffian system (an integrable connection) and a series expansion associated with the normalizing constant with an error estimation. These enable us to solve some MLE problems up to dimension $d=7$ with a specified accuracy.

math.ST↗
arXiv2013

The Holonomic Rank of the Fisher-Bingham System of Differential Equations

The Fisher-Bingham system is a system of linear partial differential equations satisfied by the Fisher-Bingham integral for the $n$-dimensional sphere $S^n$. The system is given in [Nakayama et al. (2011), Theorem 2] and it is shown that it is a holonomic system [Koyama]. We show that the holonomic rank of the system is equal to $2n+2$.

math.CA↗
arXiv2010

Computing Differential Equations for Integrals Associated to Smooth Fano Polytopes

We give an approximate algorithm of computing holonomic systems of linear differential equations for definite integrals with parameters. We show that this algorithm gives a correct answer in finite steps, but we have no general stopping condition. We apply the approximate method to find differential equations for integrals associated to smooth Fano polytopes. They are interested in the study of K3 surfaces and the toric mirror symmetry. In this class of integrals, we can apply Stienstra's rank formula to our algorithm, which gives a stopping condition of the approximate algorithm.

cs.SC↗