The monodromy conjecture for remarkable numbers

Let XX be an nn-dimensional smooth complex irreducible variety and let f:XA1f:X\to\mathbb A^1 be a regular function with non-empty zero locus. Let α\alpha be a remarkable number of ff, meaning one of the numbers defined from the combinatorics and vanishing orders associated with a log resolution of ff. Monodromy conjecture for the remarkable numbers. If α\alpha is a remarkable number of ff, then

bf(α)=0b_f(-\alpha)=0

and e2πiαe^{-2\pi i\alpha} is a monodromy eigenvalue on the nearby cycles complex of ff. This conjecture would produce roots of the bb-function and monodromy eigenvalues directly from the combinatorial remarkable numbers. The supplied text does not report a resolution.

Sources & referencesView supporting material

Primary source

Nero Budur, Eduardo de Lorenzo Poza, Quan Shi and Huaiqing Zuo, “A remarkable subset of poles of the motivic zeta function”, arXiv:2602.13508 (2026).

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