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Roots of Bernstein-Sato polynomials of certain homogeneous polynomials with two-dimensional singular loci. (English) Zbl 1457.14043
Let $$Z\subset \mathbb P^{n-1}$$, $$n\geqslant 3$$, be the hypersurface determined by a homogeneous polynomial $$f$$ of degree $$d\geqslant 3$$. Suppose that $$\dim\mathrm{Sing} Z \leqslant 1$$ (i.e. the singular locus of the affine cone $$f^{-1}(0)$$ has dimension at most 2), and that general hyperplane sections of $$Z$$ have at most weighted homogeneous isolated singularities. The second condition holds if $$n = 3$$ (in particular, $$Z$$ may be a nonreduced curve), while for $$n = 4$$ it is valid for reduced hyperplane arrangements, locally positively weighted homogeneous reduced divisors, etc.
Under these assumptions the author describes an approach for computing the roots of Bernstein-Sato polynomial supported at the origin. First he studies in detail the pole order spectral sequence (see, e.g., [A. Dimca, Singularities and topology of hypersurfaces. New York etc.: Springer-Verlag (1992; Zbl 0753.57001)]) and shows that for a given integer $$r\in [2, n]$$ this sequence degenerates at $$E_r$$-term for certain degrees. Furthermore, in the case of strongly free and locally positively weighted homogeneous divisors on $$\mathbb P^3$$ he proves that the sequence degenerates almost at $$E_2$$ and completely at $$E_3$$ and establishes a symmetry of a modified pole-order spectrum for the $$E_2$$-term (cf. [L. Narváez Macarro, Adv. Math. 281, 1242–1273 (2015; Zbl 1327.14090)]).
In fact, the author describes an algorithm for calculating the dimension of $$E_r$$-terms and the corresponding differentials, including a number of procedures implemented in algebra computer systems Macaulay, Singular and RISA/ASIR. Moreover, the section 5 contains many examples with different kinds of degeneration of the spectral sequence; all of them are examined in detail with the use of a computer similarly to [A. Dimca and G. Sticlaru, J. Symb. Comput. 91, 98–115 (2019; Zbl 1416.32014)]. The author remarks also that, in general, the computational process takes more than several hours (up to 10) to perform all necessary computations even in the case $$n = 4$$, $$d = 5$$. Finally, in the Appendix it is proved that in the case of strongly free divisors on $$\mathbb P^3$$ there is a double symmetry of the modified pole-order spectrum for the $$E_2$$-term.
##### MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 32C38 Sheaves of differential operators and their modules, $$D$$-modules 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects) 32S22 Relations with arrangements of hyperplanes
##### Software:
Macaulay2; Risa/Asir; SINGULAR
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