The Strong Monodromy Conjecture for a class of homogeneous polynomials in three variables
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908850466062336 |
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| author | Bath, Daniel Veys, Willem |
| author_facet | Bath, Daniel Veys, Willem |
| contents | We consider the class of all homogeneous, possibly non-reduced, polynomials $f$ whose associated reduced projective divisor $D_{\text{red}} \subset \mathbb{P}^{n-1}$ has (at worst) quasi-homogeneous isolated singularities. In an arbitrary number of variables $n$ and with $d$ denoting the degree of $f$, we characterize when $-n/d$ is a root of the Bernstein--Sato polynomial of $f$ in terms of elementary data involving logarithmic derivations. When we restrict to three variables, we prove the resulting class of polynomials satisfies the Strong Monodromy Conjecture, in the motivic sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20922 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Strong Monodromy Conjecture for a class of homogeneous polynomials in three variables Bath, Daniel Veys, Willem Algebraic Geometry Commutative Algebra Primary: 14F10, 14E18 We consider the class of all homogeneous, possibly non-reduced, polynomials $f$ whose associated reduced projective divisor $D_{\text{red}} \subset \mathbb{P}^{n-1}$ has (at worst) quasi-homogeneous isolated singularities. In an arbitrary number of variables $n$ and with $d$ denoting the degree of $f$, we characterize when $-n/d$ is a root of the Bernstein--Sato polynomial of $f$ in terms of elementary data involving logarithmic derivations. When we restrict to three variables, we prove the resulting class of polynomials satisfies the Strong Monodromy Conjecture, in the motivic sense. |
| title | The Strong Monodromy Conjecture for a class of homogeneous polynomials in three variables |
| topic | Algebraic Geometry Commutative Algebra Primary: 14F10, 14E18 |
| url | https://arxiv.org/abs/2602.20922 |