[Paper Review] The singular locus of Lauricella's F_C

Ryohei Hattori, Nobuki Takayama|arXiv (Cornell University)|Oct 31, 2011

Nonlinear Waves and Solitons8 references3 citations

TL;DR

This paper determines the singular locus of Lauricella's hypergeometric function $F_C$ using $D$-module theory and Gröbner basis techniques. It proves that the singular locus coincides exactly with the zero set of the polynomial $\prod_{i=1}^{m}x_{i}\prod_{\varepsilon_{i}\in\{-1,1\}}\left(1+\varepsilon_{1}\sqrt{x_{1}}+\cdots+\varepsilon_{m}\sqrt{x_{m}}\right)$, resolving a long-standing open problem in the theory of $A$-hypergeometric systems.

ABSTRACT

We determine the singular locus of the Lauricella function $F_C$ by utilizing the theory of $D$-modules and Gröbner basis. The $A$-hypergeometric system associated to $F_C$ is also discussed.

Motivation & Objective

• To rigorously determine the singular locus of the $D$-module associated with Lauricella's hypergeometric function $F_C$.
• To resolve the long-standing ambiguity in the literature regarding whether the singular locus is exactly the zero set of a certain symmetric polynomial in square roots of variables.
• To establish the singular locus of the $A$-hypergeometric system associated with $F_C$ in the complex torus using restriction and tensor product techniques.
• To confirm that the holonomic rank of the system is preserved under restriction and that the singular locus is non-empty, implying irreducibility of the monodromy representation.

Proposed method

• Utilizes $D$-module theory to analyze the characteristic variety and singular locus of the left ideal $I(m)$ generated by the differential operators annihilating $F_C$.
• Applies the $(\mathbf{0},\mathbf{1})$-initial form (principal symbol) to compute the characteristic ideal and project its zero set to the $x$-space to define the singular locus.
• Employs Gröbner basis methods in the Weyl algebra to compute the initial forms and analyze the structure of the differential operators.
• Uses the restriction algorithm to study the singular locus of the $A$-hypergeometric system in the complex torus by considering outer tensor products with $D^*z_{2m+2}^\alpha$.
• Applies the theory of $A$-hypergeometric systems, particularly the properties of holonomicity, Cohen-Macaulayness of the toric ideal, and the independence of the singular locus on the parameter $\beta$, to deduce irreducibility.
• Derives the singular locus by analyzing the determinant of a linear system in $\xi_i$, leading to the key polynomial expression involving $\varepsilon_i \in \{-1,1\}$.

Experimental results

Research questions

• RQ1 What is the exact singular locus of the $D$-module associated with Lauricella's $F_C$ function?
• RQ2 Is the singular locus of $F_C$ precisely the zero set of the symmetric polynomial $\prod_{i=1}^{m}x_i \prod_{\varepsilon_i \in \{-1,1\}} \left(1 + \sum_{j=1}^m \varepsilon_j \sqrt{x_j}\right)$?
• RQ3 How does the singular locus of the $A$-hypergeometric system associated with $F_C$ behave in the complex torus, and does it depend on the parameter $\beta$?
• RQ4 Can the irreducibility of the monodromy representation be deduced from the non-emptiness of the singular locus in the torus?
• RQ5 What is the role of the restriction algorithm and outer tensor product with $D^*z_{2m+2}^\alpha$ in computing the singular locus of the $A$-hypergeometric system?

Key findings

• The singular locus of the $D$-module $D/I(m)$ is exactly the zero set of the polynomial $\prod_{i=1}^{m}x_i \prod_{\varepsilon_i \in \{-1,1\}} \left(1 + \sum_{j=1}^m \varepsilon_j \sqrt{x_j}\right)$, which is a symmetric polynomial in the square roots of the variables.
• The proof establishes both inclusion and equality: the singular locus is contained in this variety and, by analyzing the determinant of the linear system in $\xi_i$, equality is confirmed.
• The singular locus of the $A$-hypergeometric system $H_A(\beta)$ in the complex torus is non-empty and coincides with the singular locus of the $D$-module $D/I(m)$, due to the irreducibility of the system and the Cohen-Macaulay property of the toric ideal.
• The holonomic rank of $I(m)$ is preserved under restriction and is equal to the multiplicity of the characteristic ideal at a generic point, confirming the consistency of the rank computation.
• The non-emptiness of the singular locus in the torus implies that the monodromy representation is irreducible, which supports the irreducibility of the $A$-hypergeometric system.
• The outer tensor product construction with $D^*z_{2m+2}^\alpha$ and the restriction algorithm allow the transfer of singular locus information from a higher-dimensional system to the original $F_C$ system, confirming the final result.