A system of linear partial differential equations with constant coefficients associated with ideals of finite colength (Q2565936)

The author considers an overdetermined system of partial differential equations with constant coefficients \(P_j(D) f= g_j\), \(j= 1,\dots, k\), under the standard compatibility conditions on \(g_j\). Noting \(I\) the ideal of \(\mathbb{C}[x_1,\dots, x_n]\) generated by the polynomials \(P_1,\dots, P_k\), the author assumes that \(\dim_{\mathbb{C}}\mathbb{C}[x_1,\dots, x_k]/I\) is finite. Then results are valid, which reproduce the ordinary differential case. Namely, the solutions are a vector space of finite dimension and reduce to products of polynomials by exponentials in the homogeneous case \(g_j= 0\).