某些代数超几何函数的一致化
The Uniformization of Certain Algebraic Hypergeometric Functions
摘要：
超几何函数${}_nF_{n-1}$是高阶超越函数，但对于某些参数值，它们变成代数函数，因为定义的超几何微分方程的单值性变成有限的。它表明，许多代数${}_nF_{n-1}$的，其中有限monodromy是不可约的，但imprimitive，可以表示为组合的某些明确的代数函数的一个单一的变量，即，根的三项式。这推广了Birkeland的结果，并推导出作为一个推论的一个家庭的二项式系数身份是独立的利益。三项式的根的任何元组描绘出一条射影代数曲线，并且还确定了这种所谓的Schwarz曲线何时亏格为零并且可以合理地参数化。任何这样的参数化产生一个超几何恒等式，明确一致化一个家庭的代数${}_nF_{n-1}$的。许多这样的均匀化的例子明确制定。即使当管理Schwarz曲线是正的亏格，它是如何有时可以构建明确的单值或多值参数化的个人代数${}_nF_{n-1}$的，通过参数化的一个约束Schwarz曲线。参数化要求在对称多项式的环中进行计算。
英文摘要：
The hypergeometric functions ${}_nF_{n-1}$ are higher transcendental functions, but for certain parameter values they become algebraic, because the monodromy of the defining hypergeometric differential equation becomes finite. It is shown that many algebraic ${}_nF_{n-1}$'s, for which the finite monodromy is irreducible but imprimitive, can be represented as combinations of certain explicitly algebraic functions of a single variable; namely, the roots of trinomials. This generalizes a result of Birkeland, and is derived as a corollary of a family of binomial coefficient identities that is of independent interest. Any tuple of roots of a trinomial traces out a projective algebraic curve, and it is also determined when this so-called Schwarz curve is of genus zero and can be rationally parametrized. Any such parametrization yields a hypergeometric identity that explicitly uniformizes a family of algebraic ${}_nF_{n-1}$'s. Many examples of such uniformizations are worked out explicitly. Even when the governing Schwarz curve is of positive genus, it is shown how it is sometimes possible to construct explicit single-valued or multivalued parametrizations of individual algebraic ${}_nF_{n-1}$'s, by parametrizing a quotiented Schwarz curve. The parametrization requires computations in rings of symmetric polynomials.
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