Standard bases and algebraic local cohomology for zero dimensional ideals

On efficient algorithms for computing parametric local cohomology classes associated with semi-quasihomogeneous singularities and standard bases

“…In order to compute a basis of H f 0 , we only do select terms whose weighted degrees belong to DP (d;w) . This selection of terms, is more efficient compared to our previous work [18] (nonspecial and non-parametric cases).…”

“…As CGS computation algorithm terminates, the following algorithm terminates, too. In order to introduce our new algorithm for computing a basis of H f 0 , we require the following two lemmas which are from [9,18]. Lemma 5 ([18]).…”

“…It is described in [18] how the use of algebraic local cohomology classes provides an efficient algorithm for computing (non-parametric) reduced standard bases of the Jacobi ideals. As the basic structures of parametric standard bases are determined by the structures of parametric local cohomology classes, one can generalize the algorithm in [18] to the parametric cases. The resulting algorithm also outputs a nice decomposition of the parameter spaces according to the structure of standard bases.…”

“…neous part, the list of weighted degrees of a basis of algebraic local cohomology classes is completely determined by the Poincaré polynomials associated with the weight vector [9]. In our previous works [8,18], we have introduced algorithms for computing local cohomology classes associated with the Jacobi ideals of isolated singularities. The algorithm is designed to study general hypersurface isolated singularities.…”

Computing Tjurina stratifications of $$\mu $$ μ -constant deformations via parametric local cohomology systems

“…This algorithm is the generalization of the algorithm [22] to parametric cases. Using this algorithm [14], we can also obtain a Tjurina stratification of a semi-quasihomogeneous polynomial f by giving…”