Noetherian operators and primary decomposition (Q2066948)

Let \(\mathbb{K}\) be a field of characteristic zero, \(R = \mathbb{K}[x_{1},\dots, x_{n}]\) a polynomial ring over \(\mathbb{K}\) and \(N\) a set of differential operators in the corresponding Weyl algebra \(W_{R}\), that is, in the ring of differential operators over \(R\) generated by the partial derivations \(\partial_{i} = \partial/\partial x_{i}\) (\(1\leq i\leq n\)). If \(I\) is an ideal of \(R\), then a set \(N\subseteq W_{R}\) is called a set of Noetherian operators for \(I\) if \(f\in I\) if and only if \(D\bullet f\in\sqrt{I}\) for any \(D\in N\). (\(\bullet\) denotes the action of \(D\) on \(f\) (\(x_{i}\bullet f = x_{i}f\), \(\partial_{i}\bullet f = \partial f/\partial x_{i}\)) and \(\sqrt{I}\) denotes the radical of \(I\).) Symbolic algorithms to compute Noetherian operators were developed and implemented in [\textit{A. Damiano} et al., Exp. Math. 16, No. 1, 41--53 (2007; Zbl 1136.13014); \textit{Y. Cid-Ruiz} et al., ``Primary ideals and their differential equations'', Preprint, \url{arXiv: 2001.04700}]. The paper under review addresses the problem of representing a (primary) polynomial ideal via a dual set of differential operators. In this connection, the authors present new algorithms to compute a set of Noetherian operators representing a primary ideal, as well as theoretical results leading up to them. One of the algorithms is a symbolic one; it reduces the problem to a problem of linear algebra in the spirit of the approach explored in [\textit{F. S. Macaulay}, The algebraic theory of modular systems. Cambridge: University Press (1916; JFM 46.0167.01)]. Another developed algorithm is a numerical one, it may solve problems that are out of reach for purely symbolic techniques (the authors illustrate this by a corresponding example). Given an ideal with no embedded components, this numerical algorithm combined with numerical irreducible decomposition leads to numerical primary decomposition, that is, a numerical description of all components of the ideal, which e.g. enables a probabilistic membership test.