Independence Polynomials and Hypergeometric Series

Logarithmic A-hypergeometric series II

In this paper, following [6], we continue to develop the perturbing method of constructing logarithmic series solutions to a regular A-hypergeometric system. Fixing a fake exponent of an A-hypergeometric system, we consider some spaces of linear partial differential operators with constant coefficients. Comparing these spaces, we construct a fundamental system of series solutions with the given exponent by the perturbing method. In addition, we give a sufficient condition for a given fake exponent to be an exponent. As important examples of the main results, we give fundamental systems of seri…

Characteristic cycles and Gevrey series solutions of $A$-hypergeometric systems

We compute the $L$-characteristic cycle of an $A$-hypergeometric system and higher Euler-Koszul homology modules of the toric ring. We also prove upper semicontinuity results about the multiplicities in these cycles and apply our results to analyze the behavior of Gevrey solution spaces of the system.

Differential forms on the curves associated to Appell-Lauricella hypergeometric series and the Cartier operator on them

Archinard studied the curve $C$ over $\mathbb{C}$ associated to an Appell-Lauricella hypergeometric series and differential forms on its desingularization. In this paper, firstly as a generalization of Archinard's results, we describe a partial desingularization of $C$ over a field $K$ under a mild condition on its characteristic and the space of global sections of its dualizing sheaf, especially we give an explicit basis of it. Secondly, when the characteristic is positive, we show that the Cartier operator on the space can be defined and describe it in terms of Appell-Lauricella hypergeometr…

On the integrality of hypergeometric series whose coefficients are factorial ratios

We use the Dwork-Frobenius operator to prove an integrality result for $A$-hypergeometric series whose coefficients are factorial ratios. As a special case, we generalize one direction of a classical result of Landau on the integrality of factorial ratios.

$A$-hypergeometric series associated to a lattice polytope with a unique interior lattice point

We associate to lattice points a_0,a_1,...,a_N in Z^n an A-hypergeometric series Φ(λ) with integer coefficients. If a_0 is the unique interior lattice point of the convex hull of a_1,...,a_N, then for every prime p\neq 2 the ratio Φ(λ)/Φ(λ^p) has a p-adic analytic continuation to a closed unit polydisk minus a neighborhood of a hypersurface.

Logarithm-free A-hypergeometric series

We give a dimension formula for the space of logarithm-free series solutions to an A-hypergeometric (or a GKZ hypergeometric) system. In the case where the convex hull spanned by A is a simplex, we give a rank formula for the system, characterize the exceptional set, and prove the equivalence of the Cohen-Macaulayness of the toric variety defined by A with the emptiness of the exceptional set. Furthermore we classify A-hypergeometric systems as analytic D-modules.

Hypergeometric series and Hodge cycles of four dimensional cubic hypersurfaces

In this article we find connections between the values of Gauss hypergeometric functions and the dimension of the vector space of Hodge cycles of four dimensional cubic hypersurfaces. Since the Hodge conjecture is well-known for those varieties we calculate values of Hypergeometric series on certain $CM$ points. Our methods is based on the calculation of the Picard-Fuchs equations in higher dimensions, reducing them to the Gauss equation and then applying the Abelian Subvariety Theorem to the corresponding hypergeometric abelian varieties.

$A$-hypergeometric series and a $p$-adic refinement of the Hasse-Witt matrix

We identify the $p$-adic unit roots of the zeta function of a projective hypersurface over a finite field of characteristic $p$ as the eigenvalues of a product of special values of a certain matrix of $p$-adic series. That matrix is a product $F(Λ^p)^{-1}F(Λ)$, where the entries in the matrix $F(Λ)$ are $A$-hypergeometric series with integral coefficients and $F(Λ)$ is independent of $p$.

Central charges, symplectic forms, and hypergeometric series in local mirror symmetry

We study a cohomology-valued hypergeometric series which naturally arises in the description of (local) mirror symmetry. We identify it as a central charge formula for BPS states and study its monodromy property from the viewpoint of Kontsevich's homological mirror symmetry. In the case of local mirror symmetry, we will identify a symplectic form, and will conjecture an integral and symplectic monodromy property of a relevant hypergeometric series of Gel'fand-Kapranov-Zelevinski type.

A超几何级数的希尔伯特级数与对数次数

Hilbert Series and Logarithmic Degrees of $A$-Hypergeometric Series

本文研究A超几何级数的希尔伯特级数与对数次数，构造了负支链Stanley–Reisner环的Artinian商，得到局部伪指标理想正交补等的分次维数，无需Cohen–Macaulay假设，特殊情况可化为链的h多项式。

On the $p$-integrality of $A$-hypergeometric series

Let $A$ be a set of $N$ vectors in ${\mathbb Z}^n$ and let $v$ be a vector in ${\mathbb C}^N$ that has minimal negative support for $A$. Such a vector $v$ gives rise to a formal series solution of the $A$-hypergeometric system with parameter $β= Av$. If $v$ lies in ${\mathbb Q}^n$, then this series has rational coefficients. Let $p$ be a prime number. We characterize those $v$ whose coordinates are rational, $p$-integral, and lie in the closed interval $[-1,0]$ for which the corresponding normalized series solution has $p$-integral coefficients.

一个含有m个变量、秩为p^m的超几何微分方程组

A system of hypergeometric differential equations in $m$ variables of rank $p^m$

本文构造了一类含m个变量、p+(p-1)m个参数的超几何级数及其微分方程组，统一了广义超几何级数和Lauricella F_C函数，并证明其秩为p^m、给出积分表示与奇异轨迹。