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Stratification associated with local $$b$$-functions. (English) Zbl 1184.13079
Summary: The local $$b$$-function $$b_{f,p}(s)$$ of an $$n$$-variate polynomial $$f \in \mathbb C[x](x=(x_{1},\dots ,x_n))$$ at a point $$p\in \mathbb C^n$$ is constant on each stratum of a stratification of $$\mathbb C^n$$. We propose a new method for computing such a stratification and $$b_{f,p}(s)$$ on each stratum. In the existing method proposed by T. Oaku [J. Pure Appl. Algebra 117–118, 495–518 (1997; Zbl 0918.32006)] a primary ideal decomposition of an ideal in $$\mathbb C[x,s]$$ is needed and our experiment shows that the primary decomposition can be a bottleneck for computing the stratification. In our new method, the computation can be done by just computing ideal quotients and examining inclusions of algebraic sets. The precise form of a stratum can be obtained by computing the decomposition of the radicals of the ideals in $$\mathbb C[x]$$ defining the stratum. We also introduce various techniques for improving the practical efficiency of the implementation and we show results of computations for some examples.

MSC:
 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 16S32 Rings of differential operators (associative algebraic aspects) 32C38 Sheaves of differential operators and their modules, $$D$$-modules 13N99 Differential algebra
Software:
OpenXM; Kan; Risa/Asir; SINGULAR
Full Text:
References:
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