A basis for polynomial solutions to systems of linear constant coefficient PDE's
Let \(P_k(x)\), \(k= 1, 2,\dots, m\), be polynomials in \(n\) variables with coefficients from \(K\) (\(K= \mathbb{C}\) or \(K= \mathbb{N}\)) and let \(N_{\leq M}\) be the set of all polynomial solutions of degree not greater than \(M\) to the system of partial differential equations: \(P_k(D) u(x)= 0\), \(k= 1, 2,\dots, m\) (\(m\) equations and only one unknown function). To study the set \(N_{\leq M}\) a certain associative and commutative algebra defined over \(K\) is introduced. The method of finding an easily computed (as the author claims) basis for the vector space \(N_{\leq M}\) is given. There are three illustrate examples.
scientific article; zbMATH DE number 859776