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

AF：小：线性微分方程的 A 超几何解

• 批准号: 1618657
• 负责人: Mark van Hoeij
• 金额: $ 45万
• 依托单位: Florida State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1618657&HistoricalAwards=false
• 关键词: AF; Small; Hypergeometric; Solutions; Linear

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.

许多科学定律都被捕获为微分方程——将数量（例如位置）与其导数（速度和加速度）联系起来的公式。微分方程在科学、数学和工程学中很常见。 它们可以用数值方法求解，但也可能有一个封闭式解——用熟悉的函数编写的精确解。除了小方程之外，封闭式解被认为是可以承受的。 由 PI 和他的学生开发的二阶方程算法已经证明，具有多项式系数的线性微分方程的闭式解实际上很常见；解通常可以用高斯超几何函数来编写，高斯超几何函数是微分方程中常见的函数。该项目旨在找出闭式解对于高阶方程是否也很常见。 PI 将与两名研究生合作开发算法来搜索 A 超几何函数的解决方案，该函数由 Gel'fand、Kapranov 和 Zelevinski 提出。 A 超几何函数有很多种，它们对应于多面体。 A-超几何函数可以是多元的，因此目前仅存在用于单变量方程的几种工具需要推广。 一个关键的激励问题是收敛整数级数解是否总是可以用 A 超几何函数来写。如果属实，那么 A 超几何解在涉及组合结构或积分的数学和科学领域将很常见，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。

项目成果

Sporadic Cubic Torsion

偶发立方扭转

• DOI: 10.2140/ant.2021.15.1837
• 发表时间: 2021
• 期刊: Algebra and number theory
• 影响因子: 1.3
• 作者: Derickx, Maarten;Etropolski, Anastassia;van Hoeij, Mark;Morrow, Jackson S.;Zureick-Brown, David.
• 通讯作者: Zureick-Brown, David.