Basis for power series solutions to systems of linear, constant coefficient partial differential equations
Let \(\mathbb{R}[D]=\mathbb{R}[D_1,\dots,D_n]\) denote the set of all partial differential operators with real constant coefficients in \(n\) variables and \(\mathbb{R}x\) the set of all formal power series in \(x_1,\dots,x_n\). For an ideal \(I(D)\) in \(\mathbb{R}[D]\) let \(N(I(D))=\{f\in \mathbb{R}x\mid P(D)f=0\text{ for all }P\in I(D)\}\). The author proves a representation of elements in \(N(I(D))\) which is more constructive than previously known representations of \(N(I(D))\). For this it is shown that \(\mathbb{R}[y]/I[y]\) is isomorphic to an algebra \(\mathbb{R}[\alpha]=\mathbb{R}[\alpha_1,\dots,\alpha_n]\), where the \(\alpha_i\) solve a system of polynomial relations \(I(y)=\{Q(y)\in \mathbb{R}[y]\mid Q(\alpha)=0\}\). Then a quite technical procedure involving \(X^k\otimes \alpha^k\), \(X^k=\frac{x_1^k\cdots x_n^{k_n}}{k_1!\cdots k_n!}\) yields a basis \(\{e_k(x)\mid k\in J\}\) of \(N(I(D))\). Reviewer's remark: The result is not quite that constructive because in general one will not be able to find \(\alpha\) explicitly.
scientific article; zbMATH DE number 1251311
scientific article