AF: Small: Computation of Functional Relations among Solutions of Difference and Differential Equations

地球引力对卫星产生加速度，影响卫星的速度和位置，使其轨道呈椭圆形。这只是一个数学现象的例子，它产生于物理系统的约束，可以用微分方程或差分方程来建模。该项目将开发算法，通过寻找对称性来自动揭示由微分或差分方程定义的函数的代数性质。它将产生公开可用的实现，这将对不同领域的研究人员有用，例如物理学、数论和组合学。该项目的关键组成部分包括促进与其他科学领域的互动和推进国际合作。本项目发现微分或差分方程解之间的函数关系的方法的关键是这些关系被编码在相应的伽罗瓦群的代数结构中。该项目将开发算法来确定，1)对于具有连续或离散参数的线性微分方程，其解是否满足附加的函数方程；2)对于关于移位、q扩张、马勒或椭圆曲线加法的线性差分方程，其微分方程的解是满足的；和3)对于关于q′-扩张或马勒p′-算子的线性差分方程，其解分别满足关于q′-扩张或马勒p′-算子的附加差分方程。在这三种情况中，每一种都有一个伽罗瓦理论，它编码了所寻找的解的性质。计算相应伽罗瓦群的算法直接导致计算函数关系的程序。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The pull of Earth's gravity on a satellite produces acceleration, which affects the satellite's velocity and position, making its orbit an ellipse. This is just one example of a mathematical phenomenon that arises from constraints on a physical system that can be modeled by differential or difference equations. This project will develop algorithms to automatically uncover the algebraic properties of functions defined by differential or difference equations by looking for symmetries. It will produce publicly-available implementations that will be useful to researchers in different areas, such as physics, number theory, and combinatorics. Key components of this project include the fostering of interactions with other scientific fields and the advancement of international collaborations. The key to this project's approach for discovering the functional relations among the solutions of differential or difference equations is that these relations are encoded in the algebraic structure of a corresponding Galois group. This project will develop algorithms that determine, 1) for a linear differential equation with continuous or discrete parameters, which additional functional equations are satisfied by the solutions; 2) for a linear difference equation with respect to a shift, q-dilation, Mahler, or elliptic curve addition, which differential equations are satisfied by the solutions; and 3) for a linear difference equation with respect to a q-dilation or Mahler p-operator, which additional difference equations with respect to a q'-dilation or Mahler p'-operator, respectively, are satisfied by the solutions. In each of these three situations there is a Galois theory that encodes the sought properties of the solutions. Algorithms to compute the corresponding Galois groups lead directly to procedures to compute the functional relations.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.