A new algorithm for computing logarithmic vector fields along an isolated singularity and Bruce-Roberts Milnor ideals
Let \(\mathbb{C}^{n}\) be the complex affine space of dimension \(n\) with coordinates \(\mathbf{x}=(x_1,x_2,\dots,x_{n})\). Take an open neigbourhood \(X\) of the origin \(\mathbf{0}\) of \(\mathbb{C}^{n}\) and a holomorphic function \(f(\mathbf{x})\) in \(X\) (\(f\in{\mathcal{O}}_{X}\)) such that \(S=\{\mathbf{x}\,|\,f(\mathbf{x})=0 \}\) defines a hypersurface with an isolated singularity at \(\mathbf{0}.\) A holomorphic vector field \(v\) in \(X\) is called logarithmic along \(S\) if \(v(f)\) belongs to the ideal \(\langle f \rangle\) generated by \(f\) in \({\mathcal{O}}_{X}\). Consider the sheaf \(\mathcal{Der}_{X}(-\text{log}S)\) of logarithmic vector fields along \(S\) and the module of germs \(\mathcal{Der}_{X,\mathbf{0}}(-\text{log}S)\) of \(\mathcal{Der}_{X}(-\text{log}S)\) at \(\mathbf{0}\). The main reult of the work under review stablishes a new algorithm to compute generators of \(\mathcal{Der}_{X,\mathbf{0}}(-\text{log}S)\) much faster than other algorithms introduced by the authors in a previous work [Rev. Roum. Math. Pures Appl. 64, No. 4, 523--540 (2019; Zbl 07179445)], and without using generic polar variaties in its definition, which allows to study and compute in a more efficient way generalizations of the Milnor and Tjurina numbers associated to S. Furthermore, the authors extend the algorithms to parametric cases using Gröbner systems.
scientific article; zbMATH DE number 7354277
scientific article