On polynomial solutions of PDE

TL;DR

This paper proves that certain polynomial partial differential equations have solutions within polynomial spaces of bounded degree, based on the degrees of the involved polynomials and the zero order of the differential operator.

Contribution

It establishes a degree bound for polynomial solutions to PDEs with polynomial coefficients, extending understanding of polynomial solution spaces.

Findings

Polynomial solutions exist within a degree bound related to the degrees of p and q.

The degree of solutions is at most n + s, where n is the degree of q and s is the zero order of p.

Provides a constructive bound for polynomial solutions of PDEs with polynomial coefficients.

Abstract

In this paper we prove that the PDE $p(D)f=q,$ where $p$ and $q$ are multivariate polynomials, has a solution in the space of polynomials of total degree not exceeding ${n+s},$ where $n$ is the degree of $q$ and $s$ is the zero order of $O=(0,\ldots,0)$ for $p.$