A mathematical preprint reports a proof of the multivariable Strong Monodromy Conjecture for plane curves in a set-theoretic form: every actual reduced polar hyperplane of the local multivariable topological zeta function lies in the zero locus of the multivariable Bernstein-Sato construction. In plain language, each pole hyperplane identified by the topological calculation is also picked out by the Bernstein-Sato test.
The question behind the work is whether the multivariable Strong Monodromy Conjecture holds for plane curves and how motivic and topological pole orders compare. It examines tuples of nonzero, nonunit germs vanishing at the origin, keeping the problem focused on local behavior.
The plane-curve result also allows nonreduced entries and entries with common irreducible factors, including tuples whose components are not all separate.
The match is precise
Alongside that inclusion, the preprint reports a match between two ways of recording pole data. In the plane-curve setting, the non-equivariant motivic and topological zeta functions have the same polar locus and the same generic pole orders. A polar locus is the set of parameter locations associated with poles, while pole order records the level of a pole. The result says those locations and generic orders coincide in the specified setting.
That qualifier matters. The motivic statement concerns the non-equivariant series and the multivariable Nicaise-Xu pole order. It does not cover equivariant or monodromic motivic zeta functions, and it does not settle scheme-theoretic multiplicity.
A proof built from residues and resolutions
The proof architecture combines twisted Poincare residues, resolution formulas, and specialization to one variable. The calculations use a log resolution to organize the singular data and then compare the resulting pole information with the Bernstein-Sato condition.
This machinery is not confined to the plane-curve case. In arbitrary dimension, a nonzero iterated residue class on an ordered simple-normal-crossings stratum of a log resolution gives a residue-based test: the associated negative parameter belongs to the Bernstein-Sato zero locus, and the associated rational power is not contained by the relevant differential operators. The criterion depends on both the nonzero residue class and the stated stratum conditions.
Another result covers maximal order. Every topological polar hyperplane whose order equals the ambient dimension is contained in the Bernstein-Sato zero locus. For algebraic germs, the comparison is one-sided: topological pole order is at most motivic pole order, while the two attain maximal order simultaneously. This is not a claim of equality at every order.
The boundary in higher dimensions
The higher-dimensional examples show why the plane-curve picture should not be generalized automatically. They produce an order gap in every dimension at least 3 and a difference in polar supports in every dimension at least 5. The examples mark a boundary around the plane-curve equality rather than a classification of all higher-dimensional behavior.
The main plane-curve theorem is also narrower than a full description of the Bernstein-Sato locus. It establishes set-theoretic containment for the reduced polar locus, but does not compare pole orders with scheme-theoretic multiplicities of Bernstein components or identify the remaining Bernstein-Sato components.
Taken together, the results answer the paper's central plane-curve question while keeping the scope explicit. They establish the stated topological inclusion and the matching non-equivariant motivic and topological polar data there; the residue criterion and maximal-order result extend beyond plane curves, but the constructed examples show that the full comparison does not carry over unchanged to higher dimensions.
The document identifies itself as arXiv version 3, dated 14 September 2026. It reports support from NSFC Grant No. 12601116, BJNSF Grant No. 1264052, and a Beijing municipal research program for returned overseas scholars.
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
Original paper · Full text