A mathematics preprint reports a theorem for reduced plane-curve tuples on smooth complex surfaces: every actual polar component of their local multivariable topological zeta function lies in the zero locus of the associated Bernstein–Sato ideal. In ordinary terms, every wall where the zeta function genuinely has a pole is also included among the parameter values selected by the Bernstein–Sato condition. The result addresses the topological multivariable Strong Monodromy Conjecture for this class. Its conclusion is set-theoretic: it locates the relevant components, rather than determining their multiplicities or generators.
The statement covers every tuple of reduced, non-unit plane-curve germs on a smooth complex surface germ. These are locally defined curves with no repeated component inside an entry and no entry that is merely a locally invertible function. The same irreducible branch may appear repeatedly in different entries of the tuple. This is a theorem-level mathematical result, not a study based on a recruited empirical sample.
How the argument reaches the exact wall
For surface resolutions, the proof organizes candidate polar conditions as affine hyperplanes, or “walls,” in the parameters of the zeta function. Its central reduction has two cases: a fibre-visible crossing in which two components lie on the same wall gives a pole of order two, while the no-crossing case gives a simple pole associated with a selected rupture component or the strict transform of a branch.
A central step deals with the difference between testing a wall at selected points and proving the statement for the whole affine hyperplane. The paper says that cyclic noncontainment at a Zariski-dense set of rational dual points — points spread densely in the algebraic-geometric sense — entails containment of the full affine wall in the Bernstein–Sato zero locus. That is the step that lifts pointwise rational checks to an exact-affine conclusion.
At a crossing where two components meet and both wall equations hold, a double-residue argument establishes cyclic noncontainment. More generally, the paper states that a nonzero coefficient-valued iterated residue class on an ordered simple-normal-crossings stratum is enough to entail cyclic noncontainment and place the exact corresponding Bernstein point in the Bernstein–Sato zero locus.
For the simple-pole branch involving a rupture component, the selected construction leaves at least three genuinely nonintegral residues after residue-one intersections have been filled; their quantities satisfy a balance sum of 2. The associated coefficient Poincaré residue class on the fully punctured curve is nonzero. The manuscript identifies these grouped-wall, cyclic, tube, double-residue and iterated-residue arguments as mechanisms developed in the paper, while treating Blanco’s bounds and nonvanishing results as external inputs.
What extends beyond plane curves
The paper also gives an arbitrary-dimensional result, but it is deliberately narrower. For nonzero, non-unit holomorphic germs that need not be reduced, an irreducible polar component whose generic pole order reaches the ambient dimension is contained in the Bernstein–Sato zero locus. The result does not cover lower-order poles generally, so it is not a proof of the full higher-dimensional Strong Monodromy Conjecture.
Another conclusion handles the strict-transform wall associated with every irreducible branch. That wall is contained in the Bernstein–Sato zero locus, including when a branch is repeated across entries of the tuple. The analysis treats this as a stated strict-transform input for the relevant branch vectors.
A result with clear boundaries
The main conclusion is set-theoretic and exact-affine: it places actual polar components inside the Bernstein–Sato zero locus, but does not determine scheme-theoretic multiplicities, generators or all components of that locus. The paper also does not prove a motivic version or give a complete resolution formula for the full Bernstein–Sato zero locus. In higher dimensions, its conclusion is limited to maximal-order poles and residue-based sufficient conditions.
The remaining questions are mathematical rather than empirical. They include finding geometric conditions that guarantee nonzero iterated residue classes on positive-dimensional strata, extending the argument to the full higher-dimensional conjecture, and determining whether related motivic, scheme-theoretic, threefold and toric statements can be established. The analysis also flags the challenge of distinguishing analytically varying Bernstein components from topological walls.
The document is labeled arXiv:2608.26087v1 [math.AG] and dated 26 August 2026. It is a preprint, with no journal publication reported in the supplied metadata. The acknowledgment lists support from the Beijing Natural Science Foundation, grant 1264052, and a Beijing municipal research program for returned overseas scholars. It also discloses the use of large-language-model tools for exploratory research, literature organization, drafting and internal proof checking, while stating that the author remains responsible for the manuscript.
Paper data and sources
Original title: The Multivariable Strong Monodromy Conjecture for Plane Curves
Authors: Sheng Tan
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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