AF: Small: Solving and Simplifying Algebraic, Differential, and Difference Equations.

AF：小：求解和简化代数方程、微分方程和差分方程。

• 批准号: 2007959
• 负责人: Mark van Hoeij
• 金额: $ 35万
• 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: 2020
• 资助国家: 美国
• 起止时间: 2020-10-01 至 2023-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2007959&HistoricalAwards=false
• 关键词: AF; Small; Solving; Simplifying; Algebraic

Problems in science are often reduced to mathematical equations.There are many computer programs for solving various types of equations.The equations themselves often come from computer computation, so theycan be large. This project aims to simplify large equations, reducing themwhen possible to smaller ones that can be solved more easily.When computers construct formulas and equations to tackle problems in science,their first expressions need not be the simplest ones. It is likely that simpler formulasor equations exist. Finding them makes subsequent computations easier and faster.The types of equations that this project aims to simplify are differential equations,recurrence relations, and algebraic equations in two variables. A common theme foreach of these types of equations is that in order to simplify them, a key step is atechnical analysis of their singularities.Computer algebra systems can automatically find exact solutions for various typesof equations. They are an important part of the infrastructure for researchand education. However, they often produce unnecessarily complicated answers.The techniques developed in this project will help to simplify not only equations,but solutions as well, making them more useful for research and teaching.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

科学中的问题常常被简化为数学方程。有许多计算机程序可以解各种类型的方程。方程本身常常来自计算机计算，所以它们可能很大。这个项目旨在简化大型方程，尽可能将它们简化为更容易求解的小方程。当计算机构造公式和方程来解决科学问题时，它们的第一个表达式不需要是最简单的。很可能存在更简单的公式或方程。找到它们可以使后续的计算更容易和更快。本项目旨在简化的方程类型是微分方程、递归关系和二元代数方程。这些类型的方程的一个共同点是，为了简化它们，关键的一步是对它们的奇点进行技术分析。计算机代数系统可以自动找到各种类型方程的精确解。它们是研究和教育基础设施的重要组成部分。然而，它们往往会产生不必要的复杂答案。该项目开发的技术不仅有助于简化方程，还有助于简化解决方案，使其对研究和教学更有用。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

项目成果

Desingularization and p-Curvature of Recurrence Operators

递归算子的去奇异化和 p 曲率

• DOI: 10.1145/3476446.3535478
• 发表时间: 2022
• 期刊: ISSAC'2022
• 作者: Zhou, Yi;van Hoeij, Mark
• 通讯作者: van Hoeij, Mark

A Divisor Formula and a Bound on the $$\mathbb {Q}$$-Gonality of the Modular Curve $$X_1(N)$$

除数公式和模曲线 $$X_1(N)$$ 的 $$mathbb {Q}$$-Gonality 上的界限

• DOI: 10.1007/s40993-021-00243-3
• 发表时间: 2021
• 期刊: Research in Number Theory
• 影响因子: 0.8
• 作者: van Hoeij, Mark;Smith, Hanson
• 通讯作者: Smith, Hanson