[Paper Review] Modified A-hypergeometric Systems

Nobuki Takayama|ArXiv.org|Jun 30, 2007

Polynomial and algebraic computation9 references3 citations

TL;DR

This paper introduces a modified ${\cal A}$-hypergeometric system by extending the original GKZ system with an additional variable $t$, using a change of variables inspired by Gröbner deformation and blowing-up geometry. It establishes the holonomicity and rank equality with the original system, and derives the indicial polynomial along $t=0$ as a key tool for analyzing local solutions, proving that the rank of $H_A(\beta)$ is at least ${\rm vol}(A)$ for non-homogeneous $A$. The modified system enables new insights into slopes and formal series solutions, particularly when the indicial polynomial vanishes or has non-integral root differences.

ABSTRACT

We will introduce a modified system of A-hypergeometric system (GKZ system) by applying a change of variables for Groebner deformations and study its Groebner basis and the indicial polynomials along the "exceptional hypersurface".

Motivation & Objective

• To introduce a modified ${\cal A}$-hypergeometric system by extending the original GKZ system with an auxiliary variable $t$ to facilitate local analysis near $t=0$.
• To study the Gröbner basis and indicial polynomial of the modified system along the hypersurface $t=0$ as a first step toward global and local analysis.
• To provide a new proof of the lower bound $\text{rank}(H_A(\beta)) \geq \text{vol}(A)$ for non-homogeneous $A$, resolving a question posed by Go Okuyama.

Proposed method

• The modified system $H_{A,w}(\beta)$ is defined on $({\bf C}^n \times {\bf C}^*)$ via a change of variables $y_i = t^{w_i}x_i$, lifting the original ${\cal A}$-hypergeometric system on $y$-space.
• The system is constructed as a left ideal in the $D$-module ring $D = {\bf C}\langle x_1,\dots,x_n,t,\partial_1,\dots,\partial_n,\partial_t\rangle$, incorporating differential operators and toric relations from the extended matrix $\tilde{A}$.
• Gröbner basis techniques are applied to the toric ideal $I_{\tilde{A}}$ in ${\bf C}[\partial_1,\dots,\partial_n,t]$, with saturation used to ensure $t^m \notin I_{\tilde{A}}$.
• The indicial polynomial along $t=0$ is derived using standard pairs and distraction of the initial ideal $\text{in}_\tau(I_{\tilde{A}})$, yielding $\sum_{(\partial^\beta,\sigma) \in \mathcal{T}(M)} (s - w \cdot \beta^{(\partial^\beta,\sigma)})$.
• Laplace transformation with respect to $t$ is used to relate the modified system to the original ${\cal A}$-hypergeometric system for $\tilde{A}$, proving holonomicity and rank preservation.
• Formal series solutions of the form $t^e \sum c_k(x) t^k$ are constructed when the indicial polynomial is non-zero and root differences are non-integral.

Experimental results

Research questions

• RQ1 How can the ${\cal A}$-hypergeometric system be modified to better analyze local behavior near $t=0$, particularly in relation to slopes and monodromy?
• RQ2 What is the structure of the indicial polynomial of the modified system along the hypersurface $t=0$, and how does it relate to the existence of formal solutions?
• RQ3 Can the modified system provide a new proof of the lower bound $\text{rank}(H_A(\beta)) \geq \text{vol}(A)$ for non-homogeneous $A$?
• RQ4 Under what conditions does the modified system admit formal Puiseux series solutions along $t=0$, and when does it not?

Key findings

• The modified system $H_{A,w}(\beta)$ is holonomic and has the same holonomic rank as the original $H_A(\beta)$, regardless of the choice of $w \in \mathbb{Z}^n$.
• The indicial polynomial along $t=0$ is given by $\sum_{(\partial^\beta,\sigma) \in \mathcal{T}(M)} (s - w \cdot \beta^{(\partial^\beta,\sigma)})$, where $M = \text{in}_\tau(I_{\tilde{A}})$, and is zero if $\mathcal{T}(M)$ is empty.
• When the indicial polynomial is non-zero and has non-integral root differences, the system admits formal series solutions of the form $t^e \sum_{k=0}^\infty c_k(x) t^k$ with $c_k \in {\bf C}[1/x_1,\dots,1/x_n,x_1,\dots,x_n]$.
• For $A = (-1,1,2)$, $\beta = (1/2)$, $w = (-2,-1,0)$, the indicial polynomial is zero, indicating no such formal solutions exist, consistent with the system being related to the Bessel function.
• When $w = (3,2,1)$, the indicial polynomial is $(s - 3/2)(s + 1/4)(s - 7/4)$, computed via Risa/Asir's `generic_bfct`, confirming the formula in Theorem 3.
• The local monodromy group around $t=0$ is generated by $\text{diag}(-1, \exp(\pi i /2), -\exp(\pi i /2))$, indicating non-trivial monodromy behavior.