Hilbert series of graded Milnor algebras and roots of Bernstein-Sato polynomials

Morihiko Saito

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Hilbert series of graded Milnor algebras and roots of Bernstein-Sato polynomials

Morihiko Saito

Algebraic Geometry · Submitted 21 Sep 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This shows that the roots of the Bernstein-Sato polynomials strictly supported at the origin cannot be determined uniquely by the Hilbert series of the Milnor algebras.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,288 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.54 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.90 · mentions code or data availability 0.06.

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arXiv listing abstract v7. Results and the main claim are underlined; show every role.

We show that there is a pair of homogeneous polynomials such that the sets of roots of their Bernstein-Sato polynomials which are strictly supported at the origin are different although the sets of roots which are not strictly supported at the origin are the same and moreover their graded Milnor algebras have the same Hilbert series. This shows that the roots of the Bernstein-Sato polynomials strictly supported at the origin cannot be determined uniquely by the Hilbert series of the Milnor algebras. This is contrary to certain hyperplane arrangement cases. It also implies that a nonzero torsion element with pure degree in the Milnor algebra does not necessarily contribute to a root of the Bernstein-Sato polynomial in an expected way. This example is found by using Macaulay2 and RISA/ASIR.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We show that there is a pair of homogeneous polynomials such that the sets of roots of their Bernstein-Sato polynomials which are strictly supported at the origin are different although the sets of roots which are not strictly supported at the origin are the same and moreover their graded Milnor algebras have the same Hilbert series.	result 0.95	0.96	0.71	asserted 1.00
s1	This shows that the roots of the Bernstein-Sato polynomials strictly supported at the origin cannot be determined uniquely by the Hilbert series of the Milnor algebras.	interpretation 0.91	0.91	0.89	asserted 1.00
s2	This is contrary to certain hyperplane arrangement cases.	interpretation 0.88	0.13	0.05	asserted 1.00
s3	It also implies that a nonzero torsion element with pure degree in the Milnor algebra does not necessarily contribute to a root of the Bernstein-Sato polynomial in an expected way.	interpretation 0.98	0.75	0.08	asserted 0.63
s4	This example is found by using Macaulay2 and RISA/ASIR.	method 1.00	0.93	0.02	asserted 1.00

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Original abstract

We show that there is a pair of homogeneous polynomials such that the sets of roots of their Bernstein-Sato polynomials which are strictly supported at the origin are different although the sets of roots which are not strictly supported at the origin are the same and moreover their graded Milnor algebras have the same Hilbert series. This shows that the roots of the Bernstein-Sato polynomials strictly supported at the origin cannot be determined uniquely by the Hilbert series of the Milnor algebras. This is contrary to certain hyperplane arrangement cases. It also implies that a nonzero torsion element with pure degree in the Milnor algebra does not necessarily contribute to a root of the Bernstein-Sato polynomial in an expected way. This example is found by using Macaulay2 and RISA/ASIR.

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6 arXiv papers cite this in 13 sentences

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Compares with it 2 papers

He has produced a similar proof that is based on cyclic covers rather than the closure of the Milnor fiber in $^n$, see the appendix of [ref] . * If the natural map [ref] is an isomorphism, one obtains a natural map $[H^0_(R_n/(f))]_d-n+i^Hodge_n-2(H^n-1(M_f,)_λ)$.

The Jacobian module, the Milnor fiber, and the $D$-module generated by $f^s$ · Milnor fiber and Jacobian module · 0.90
Our algorithm shows that in both cases $H^1(F,)=0$, $q_0(f_1)=9$, $q_0(f_2)=8$, while the spectrum on $H^2(F,)$ is given by [equation] and, respectively, [equation] hence exactly the same formulas as in [ref] , equations (1.4.5).

Computing the monodromy and pole order filtration on Milnor fiber cohomology of plane curves · Examples · 0.98

Cites it as support 1 paper

* The zero set $R_f$ is not determined by the Hilbert function for its Milnor algebra $M(f)$ for a reduced plane curve $C:f=0$, see [ref] and Example [ref] below.

Computing the monodromy and pole order filtration on Milnor fiber cohomology of plane curves · Application to the study of Bernstein-Sato polynomials · 0.97

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