Roots of Bernstein-Sato polynomials of certain homogeneous polynomials with two-dimensional singular loci.

*(English)*Zbl 1457.14043Let \(Z\subset \mathbb P^{n-1}\), \(n\geqslant 3\), 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. NarvĂˇez 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.

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. NarvĂˇez 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.

Reviewer: Aleksandr G. Aleksandrov (Moskva)

##### 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 |