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

Abstract

Algebraic local cohomology classes associated with parametric semiquasihomogeneous hypersurface isolated singularities are considered in the context of symbolic computation. The motivations for this paper are computer calculations of complete lists of Tjurina numbers of semi-quasihomogeneous polynomials with isolated singularity. A new algorithm, that utilizes parametric local cohomology systems, is proposed to compute Tjurina stratifications associated with μ-constant deformations of weighted homogeneous isolated singularities. The resulting algorithm gives in particular a suitable decomposition of the parameter space depending on the structure of the parametric local cohomology systems. An efficient algorithm of computing parametric standard bases of relevant ideals is also given as an application of parametric local cohomology systems.

Computing Grothendieck Point Residues via Solving Holonomic Systems of First Order Partial Differential Equations

“…In order to represent local cohomology classes in H m [O] (K[z]), we adopt the notations introduced in [19,20].…”

Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues

“…The method above computes a basis of non-trivial logarithmic vector fields. Each step can be effectively executable, as in [40], by utilizing algorithms described in [20,21,22,41].…”

Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues

“…Therefore, the proposed method can be used as a basic procedure for computing Gauss-Manin connection. Each step can be effectively executable, as in [33], by utilizing algorithms described in [18,19,20,32]. One of the advantage of the proposed method lies in the fact that the resulting algorithm can handle parametric cases.…”

“…The weighted degree of the upper monomial txy 6 is equal to 25, Accordingly f is a quasi homogeneous function. In fact, by using an algorithm described in [18,31], we find that f is in the ideal ( ∂f ∂x , ∂f ∂y ). Therefore, by a classical result of K. Saito ( [24]), f is quasi-homogeneous.…”