Motivic, multiplicity, higher-dimensional, and Bernstein-locus extensions

Establish the motivic analogue of the Topological Multivariable Strong Monodromy Conjecture, determine scheme-theoretic multiplicities of Bernstein components, prove the full higher-dimensional Strong Monodromy Conjecture, and derive a resolution formula for the whole Bernstein–Sato zero locus Z(B_F).

Background

The paper proves the Topological Multivariable Strong Monodromy Conjecture for tuples of reduced plane curve germs and proves, in arbitrary dimension, only the implication from a nonzero coefficient-valued iterated residue class to Bernstein–Sato vanishing, together with containment for maximal-order polar hyperplanes. The authors explicitly identify several broader extensions that are not resolved by their arguments.

These unresolved directions concern motivic rather than merely topological zeta functions, scheme-theoretic information rather than set-theoretic containment, the full strong conjecture in dimensions beyond surfaces, and a resolution-theoretic description of all components of the multivariable Bernstein–Sato zero locus.

References

Several structurally different problems remain open. The argument does not prove the motivic conjecture, scheme-theoretic multiplicities of Bernstein components, the full higher-dimensional Strong Monodromy Conjecture, or a resolution formula for the whole of $Z(B_F)$.

— The Multivariable Strong Monodromy Conjecture for Plane Curves  (2608.26087 - Tan, 26 Aug 2026) in Section 8, “Further questions”

The central geometric question is therefore which conditions force the iterated residue class on a positive-dimensional SNC stratum to be nonzero. Uniform answers for threefold curve strata, toric resolutions, and Newton-nondegenerate tuples would require additional geometry and are not claimed here.

— The Multivariable Strong Monodromy Conjecture for Plane Curves  (2608.26087 - Tan, 26 Aug 2026) in Section 8, “Further questions”

In one variable, the Bernstein polynomial can vary in equisingular families Example~1. Thus even a decorated resolution graph cannot by itself determine the entire Bernstein--Sato ideal; the boundary between topological walls and analytically varying components remains to be understood.

— The Multivariable Strong Monodromy Conjecture for Plane Curves  (2608.26087 - Tan, 26 Aug 2026) in Section 8, “Further questions”

Now we can state the monodromy conjecture. Let X be a smooth complex algebraic variety and let f\in\Gamma(X,\mathcal O_X) be a nonconstant regular function. If s_0 is a pole of Z_{f}{\mathrm{mot}}(s), then e{s_0\cdot 2\pi \sqrt{-1}}\in E(f).

— A Polynomial PDE Criterion for the $-n/d$ Root of the Bernstein-Sato polynomial of Homogeneous Ideals  (2609.03976 - Chen et al., 3 Sep 2026) in Section 2, subsection “Motivic Zeta Functions and the Monodromy Conjecture,” Conjecture labeled “Monodromy Conjecture” (\ref{mc})

However, the strong monodromy conjecture is still open in the general case.

— Uniform estimate of Lang-Weil type for counting points of schemes over finite rings  (2610.01748 - Nguyen, 1 Oct 2026) in Section 1, Introduction