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Polynomial solutions to constant coefficient differential equations


Author: S. Paul Smith
Journal: Trans. Amer. Math. Soc. 329 (1992), 551-569
MSC: Primary 35E20; Secondary 35C05
DOI: https://doi.org/10.1090/S0002-9947-1992-1013339-6
MathSciNet review: 1013339
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Abstract: Let $ {D_1}, \ldots ,{D_r} \in \mathbb{C}[\partial /\partial {x_1}, \ldots ,\partial /\partial {x_n}]$ be constant coefficient differential operators with zero constant term. Let

$\displaystyle S = \{ f \in \mathbb{C}[{x_1}, \ldots ,{x_n}]\vert{D_j}(f) = 0\;{\text{for all }}1 \leqslant j \leqslant r\} $

be the space of polynomial solutions to the system of simultaneous differential equations $ {D_j}(f) = 0$. It is proved that $ S$ is a module over $ \mathcal{D}(V)$, the ring of differential operators on the affine scheme $ V$ with coordinate ring $ \mathbb{C}[\partial /\partial {x_1}, \ldots ,\partial /\partial {x_n}]/\left\langle {{D_1}, \ldots ,{D_r}} \right\rangle $. If $ V$ is smooth and irreducible, then $ S$ is a simple $ \mathcal{D}(V)$-module, $ S = 1.\mathcal{D}(V)$, and the generators for $ \mathcal{D}(V)$ yield an algorithm for obtaining a basis for $ S$. If $ V$ is singular, then $ S$ need not be simple. However, $ S$ is still a simple $ \mathcal{D}(V)$-module for certain curves $ V$, and certain homogeneous spaces $ V$, and this allows one to obtain a basis for $ S$, through knowledge of $ \mathcal{D}(V)$.

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DOI: https://doi.org/10.1090/S0002-9947-1992-1013339-6
Article copyright: © Copyright 1992 American Mathematical Society

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