AF: Small: A-Hypergeometric Solutions of Linear Differential Equations

许多科学定律被归结为微分方程式--将一个量(如位置)与其导数(速度和加速度)联系起来的公式。微分方程式在科学、数学和工程中都很常见。它们可以用数值求解，但也可能有一个闭合形式的解--用熟悉的函数写成的精确解。除了小的方程外，人们认为是闭合形式的解。由PI和他的学生开发的二阶方程的算法已经证明，具有多项式系数的线性微分方程组的闭合形式解实际上是常见的；解通常可以写成高斯超几何函数，这是微分方程式中常见的函数。这个项目的目的是找出闭合形式解是否也适用于高阶方程。PI将与两名毕业生合作，开发算法来搜索由Gel‘fand，Kapranov和Zlevinski引入的超几何函数的解。有各种各样的A超几何函数，它们对应于多面体。A-超几何函数可以是二元的，所以目前只存在于一元方程的几个工具需要推广。一个关键的动机问题是收敛的整数级数解是否总是可以用A-超几何函数来表示。如果是真的，那么超几何解将在涉及组合结构或积分的数学和科学领域中很常见，例如物理学中的伊辛模型或费曼图。这个问题引出了其他几个A-超几何函数，其中一些可以通过本项目中开发的算法来解决。

Many scientific laws are captured as differential equations --formulas that relates a quantity (such as position) to its derivatives(velocity and acceleration). Differential equations are common inscience, mathematics and engineering. They can be solved numerically,but may also have a closed-form solution -- an exact solution writtenin terms of familiar functions.Except for small equations, closed-form solutions were thought to berare. Algorithms for second-order equations, which were developed bythe PI and his students, have demonstrated that closed form solutions of linear differential equations with polynomial coefficientsare actually common; solutions can often be written in terms ofGauss's hypergeometric function, a familiar function in differentialequations.This project aims to find out if closed form solutions are also commonfor higher order equations. The PI will work with two graduatestudents to develop algorithms to search for solutions in terms ofA-hypergeometric functions, which were introduced by Gel'fand,Kapranov and Zelevinski. There is a wide variety of A-hypergeometric functions, whichcorrespond to polytopes. A-hypergeometric functions can bemultivariate, so several tools that currently only exist forunivariate equations need to be generalized. A key motivatingquestion is if convergent integer-series solutions can always bewritten in terms of A-hypergeometric functions. If true, thenA-hypergeometric solutions would be common in areas of mathematics andscience that involve combinatorial structures or integrals, such as the Ising modelor Feynman diagrams in physics. This question leads to several otherson A-hypergeometric functions, some of which can be settled by thealgorithms to be developed in this project.