arxiv: 2605.07077 · v1 · submitted 2026-05-08 · 🧮 math.CO

Recognition: 2 theorem links

· Lean Theorem

Topological Zeta Functions of Matroids: Operations and Computations

Dawit Mengesha , Robert Miranda , Brian Sun

Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:31 UTC · model grok-4.3

classification 🧮 math.CO MSC 05B35

keywords topological zeta functionmatroidtruncationfree extensionMöbius inversiongirthTaylor coefficientsvaluative invariant

0 comments

The pith

Topological zeta functions of matroids satisfy a recurrence relation under truncation and free extension, with Taylor coefficients fixed by the girth.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the topological zeta function of a matroid, treated as a valuative invariant on the lattice of flats, obeys a recurrence derived from Möbius inversion. This recurrence expresses the zeta function of the truncation and of the free extension directly in terms of the zeta function of the original matroid. The Taylor coefficients of the zeta function are further identified with a single matroid invariant called the girth. These results supply explicit computational rules and extend an earlier characterization of the same coefficients.

Core claim

The topological zeta function is a rational function and valuative invariant of a matroid. A clean recurrence for the Möbius inversion on the lattice of flats yields explicit formulas for the zeta function of the truncation and the free extension in terms of the zeta function of the given matroid. The Taylor coefficients of any such zeta function are completely determined by the girth of the matroid, generalizing the earlier result of Jensen-Kutler-Usatine.

What carries the argument

The recurrence relation obtained by applying Möbius inversion to the topological zeta function on the flat lattice, which converts the effect of truncation and free extension into an algebraic relation between rational functions.

If this is right

The zeta function of any truncated matroid is obtained from the original by a fixed algebraic transformation given by the recurrence.
The same transformation gives the zeta function of the free extension.
All Taylor coefficients of the zeta function depend only on the girth and are therefore independent of other details of the matroid.
The same coefficient characterization applies to every matroid, extending the special case treated by Jensen-Kutler-Usatine.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The recurrence may be iterated to obtain zeta functions for sequences of successive truncations without recomputing the full lattice each time.
Because girth is a circuit-size invariant, the coefficient formula links the zeta function to the matroid's circuit structure in a direct way.
Similar recurrences could be sought for other standard matroid operations such as deletion and contraction.

Load-bearing premise

The topological zeta function remains a valuative invariant under truncation, free extension, and the associated lattice operations for every matroid.

What would settle it

Explicit computation of the topological zeta function for the uniform matroid U_{2,4} and for its truncation U_{1,4}, followed by direct verification that the rational functions satisfy the stated recurrence.

read the original abstract

The topological zeta function of a matroid is a rational function as well as a valuative invariant of the matroid, encoding rich combinatorial information. We analyze topological zeta functions of matroids from the vantage point of several matroid operations and operations on lattices of flats. We prove a clean recurrence relation for the M\"obius inversion and use it to describe the topological zeta function of the truncation and free extension of a matroid in relation with that of the original matroid. We also characterize the Taylor coefficients of the topological zeta functions for matroids in terms of a matroid invariant, which we call the girth, and generalize an earlier result by Jensen-Kutler-Usatine.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper supplies explicit recurrences for the topological zeta function under truncation and free extension, plus a girth characterization of the Taylor coefficients that generalizes prior work.

read the letter

The main advance here is a pair of recurrence relations that let you compute the topological zeta function of a truncated matroid or its free extension from the original one, along with a description of the Taylor coefficients in terms of the girth invariant. These build directly on the valuative property and Möbius inversion over the lattice of flats. The derivations stay within standard matroid operations and do not add extra restrictions on rank or simplicity. The generalization of the Jensen-Kutler-Usatine result follows from the same setup and looks like a usable extension rather than a restatement. The manuscript states the relations cleanly and supplies the needed definitions, so the claims are checkable from the text. No circularity or hidden fitting appears in the argument. A small presentational gap is the limited number of worked examples, but that does not affect the logic of the recurrences themselves. The paper stays focused on computation within matroid theory and does not claim broader reach. Specialists already working with topological zeta functions or lattice invariants will find the formulas directly applicable for extending calculations to new families. Readers outside this narrow area will see little immediate use. The work shows clear engagement with the existing literature and produces concrete, falsifiable relations. It deserves a serious referee because the methods are grounded and the results are explicit enough to be tested on examples.

Referee Report

0 major / 2 minor

Summary. The manuscript proves a recurrence relation for the Möbius inversion of the topological zeta function of a matroid, derives explicit relations expressing the topological zeta functions of the truncation and free extension in terms of the original matroid, and characterizes the Taylor coefficients of these zeta functions via a new matroid invariant called the girth, thereby generalizing an earlier result of Jensen-Kutler-Usatine. The arguments rely on the valuative property of the zeta function together with direct manipulation of the lattice of flats.

Significance. If the derivations hold, the work supplies practical operational tools for computing and relating topological zeta functions under standard matroid operations and introduces the girth invariant as a means to extract Taylor coefficients. These contributions extend the framework of Jensen-Kutler-Usatine and may facilitate explicit calculations in matroid theory.

minor comments (2)

[Abstract] The abstract states that proofs of the recurrence and the girth characterization are given, yet the manuscript would benefit from one or two fully worked small-matroid examples (e.g., uniform matroids of small rank) that verify both the recurrence and the coefficient formula numerically.
The introduction should explicitly recall the definition of the topological zeta function (including its expression via the Möbius function on the lattice of flats) before stating the new recurrence, to make the subsequent derivations self-contained for readers unfamiliar with the Jensen-Kutler-Usatine setup.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of our manuscript on topological zeta functions of matroids, which correctly identifies the recurrence relation for the Möbius inversion, the explicit formulas relating the zeta functions of truncations and free extensions to the original matroid, and the characterization of Taylor coefficients via the new girth invariant (generalizing Jensen-Kutler-Usatine). We appreciate the noted significance of these operational tools and the recommendation for minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The central results consist of a recurrence for the Möbius function applied to the topological zeta function, explicit transformation rules under truncation and free extension, and a girth-based description of Taylor coefficients that generalizes an independent prior result of Jensen-Kutler-Usatine. These steps rest on the standard definition of the zeta function as a valuative invariant together with direct algebraic manipulation of the lattice of flats and the Möbius inversion formula, both of which are externally defined and do not presuppose the paper's conclusions. No parameter is fitted to data and then renamed as a prediction, no ansatz is imported via self-citation, and the cited prior work is by different authors. The derivation chain therefore remains non-circular and externally verifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claims rest on the topological zeta function being a valuative invariant and on standard properties of Möbius inversion in the lattice of flats; girth is introduced as a new named invariant without independent evidence supplied in the abstract.

axioms (2)

standard math Möbius inversion formula applies directly to the lattice of flats of a matroid
Invoked to obtain the clean recurrence relation for the zeta function under operations.
domain assumption Topological zeta function is a valuative invariant
Stated in the abstract as the starting point for all subsequent analysis and characterizations.

invented entities (1)

girth matroid invariant no independent evidence
purpose: Characterize the Taylor coefficients of the topological zeta function
Newly named invariant used to express the coefficients; no independent evidence or prior definition supplied.

pith-pipeline@v0.9.0 · 5408 in / 1452 out tokens · 20532 ms · 2026-05-11T01:31:07.940891+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.2 recurrence for Y_M(s) = -1/|E|s+rk(M) sum ... ; Theorem 5.2 girth g Taylor coeffs (-1)^k |E|_k for k < g
IndisputableMonolith/Foundation/AlexanderDuality alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 2.1 topological zeta via flags and characteristic polynomials on degeneration MF

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

51 extracted references · 51 canonical work pages

[1]

Journal of the London Mathematical Society , volume=

The motivic zeta functions of a matroid , author=. Journal of the London Mathematical Society , volume=. 2021 , publisher=

work page 2021
[2]

Denef, Jan and Loeser, Fran. Caract. Journal of the American Mathematical Society , pages=. 1992 , publisher=

work page 1992
[3]

Selecta Mathematica , volume=

Incidence combinatorics of resolutions , author=. Selecta Mathematica , volume=. 2004 , publisher=

work page 2004
[4]

2010 , publisher=

Matroid theory , author=. 2010 , publisher=

work page 2010
[5]

Discrete Mathematics , volume=

Combinatorial analogs of topological zeta functions , author=. Discrete Mathematics , volume=. 2019 , publisher=

work page 2019
[6]

Combinatorial Geometries , DOI=

Neil White , place=. Combinatorial Geometries , DOI=. 1987 , collection=

work page 1987
[7]

The Monodromy Conjecture for hyperplane arrangements , volume =

Budur, Nero and Mustaţă, Mircea and Teitler, Zach , year =. The Monodromy Conjecture for hyperplane arrangements , volume =. Geometriae Dedicata , doi =

work page
[8]

and Knuth, Donald E

Graham, Ronald L. and Knuth, Donald E. and Patashnik, Oren , title =. 1994 , isbn =

work page 1994
[9]

An Introduction to Chromatic Polynomials , volume=

Read, Ronald , year=. An Introduction to Chromatic Polynomials , volume=. Journal of Combinatorial Theory , publisher=. doi:10.1016/S0021-9800(68)80087-0 , number=

work page doi:10.1016/s0021-9800(68)80087-0
[10]

1979 , publisher =

Combinatorial Theory , author =. 1979 , publisher =

work page 1979
[11]

The Möbius Function and the Characteristic Polynomial , DOI=

Zaslavsky, Thomas , year=. The Möbius Function and the Characteristic Polynomial , DOI=. Combinatorial Geometries , publisher=

work page
[12]

2012 , publisher =

Young Tableaux , author =. 2012 , publisher =

work page 2012
[13]

Publications Math\'ematiques de l'IH\'ES , publisher =

Grothendieck, Alexander , title =. Publications Math\'ematiques de l'IH\'ES , publisher =. 1964 , pages =

work page 1964
[14]

Families of varieties of general type , author=

Families of varieties of general type , url=. Families of varieties of general type , author=

work page
[15]

2000 , publisher=

Singular loci of Schubert varieties , author=. 2000 , publisher=

work page 2000
[16]

1977 , publisher =

Algebraic Geometry , author =. 1977 , publisher =

work page 1977
[17]

Linear Algebraic Groups and Finite Groups of Lie Type , DOI=

Malle, Gunter and Testerman, Donna , year=. Linear Algebraic Groups and Finite Groups of Lie Type , DOI=

work page
[18]

1963 , publisher=

Foundations of differential geometry , author=. 1963 , publisher=

work page 1963
[19]

Singularities of Generalized Richardson Varieties , volume =

Billey, Sara and Coskun, Izzet , year =. Singularities of Generalized Richardson Varieties , volume =. Communications in Algebra , doi =

work page
[20]

Lectures on the Geometry of Flag Varieties , bookTitle=

Brion, Michel , editor=. Lectures on the Geometry of Flag Varieties , bookTitle=. 2005 , publisher=

work page 2005
[21]

Smoothness of Schubert varieties via patterns in root subsystems

Sara Billey and Alexander Postnikov. Smoothness of Schubert varieties via patterns in root subsystems. Advances in Applied Mathematics. 2005. doi:https://doi.org/10.1016/j.aam.2004.08.003

work page doi:10.1016/j.aam.2004.08.003 2005
[22]

Applications of the Theory of Morse to Symmetric Spaces , volume =

Raoul Bott and Hans Samelson , journal =. Applications of the Theory of Morse to Symmetric Spaces , volume =

work page
[23]

Flags, Schubert polynomials, degeneracy loci, and determinantal formulas

Fulton, William. Flags, Schubert polynomials, degeneracy loci, and determinantal formulas. Duke Math. J. 1992. doi:10.1215/S0012-7094-92-06516-1

work page doi:10.1215/s0012-7094-92-06516-1 1992
[24]

Abe, Hiraku and Billey, Sara , year =

work page
[25]

and Sandhya, B

Lakshmibai, V. and Sandhya, B. Criterion for smoothness of Schubert varieties in Sl(n)/B. Proceedings of the Indian Academy of Sciences - Mathematical Sciences. 1990. doi:10.1007/BF02881113

work page doi:10.1007/bf02881113 1990
[26]

and Reiner, V

Gasharov, V. and Reiner, V. , title =. Journal of the London Mathematical Society , volume =. doi:10.1112/S0024610702003605 , url =. https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/S0024610702003605 , abstract =

work page doi:10.1112/s0024610702003605
[27]

Functional Analysis and Its Applications , volume=

Small resolutions of singularities of Schubert varieties , author=. Functional Analysis and Its Applications , volume=. 1983 , publisher=

work page 1983
[28]

Annales scientifiques de l'\'Ecole Normale Sup\'erieure , publisher =

Demazure, Michel , title =. Annales scientifiques de l'\'Ecole Normale Sup\'erieure , publisher =. 1974 , pages =. doi:10.24033/asens.1261 , zbl =

work page doi:10.24033/asens.1261 1974
[29]

H. C. Hansen , journal =. ON CYCLES IN FLAG MANIFOLDS , volume =

work page
[30]

Journal of Differential Geometry , volume=

Rigid and non-smoothable Schubert classes , author=. Journal of Differential Geometry , volume=. 2011 , publisher=

work page 2011
[31]

2018 , journal=

The Gieseker-Petri theorem and imposed ramification , author=. 2018 , journal=

work page 2018
[32]

2019 , eprint=

Relative Richardson Varieties , author=. 2019 , eprint=

work page 2019
[33]

Families of Varieties of General Type: the Shafarevich Conjecture and Related Problems

Kov \'a cs, S \'a ndor J. Families of Varieties of General Type: the Shafarevich Conjecture and Related Problems. Higher Dimensional Varieties and Rational Points. 2003. doi:10.1007/978-3-662-05123-8_6

work page doi:10.1007/978-3-662-05123-8_6 2003
[34]

Singularities of richardson varieties

Allen Knutson and Alexander Woo and Yong, Alexander Ng Tengfat. Singularities of richardson varieties. Mathematical Research Letters. 2013. doi:10.4310/MRL.2013.v20.n2.a14

work page doi:10.4310/mrl.2013.v20.n2.a14 2013
[35]

Transactions of the American Mathematical Society , volume=

A Formula for the Tangent Bundle of Flag Manifolds and Related Manifolds , author=. Transactions of the American Mathematical Society , volume=. 1975 , publisher=

work page 1975
[36]

Stacks for Everybody

Fantechi, Barbara. Stacks for Everybody. European Congress of Mathematics. 2001

work page 2001
[37]

2014 , eprint=

Matroid theory for algebraic geometers , author=. 2014 , eprint=

work page 2014
[38]

Geometric combinatorics , volume=

An introduction to hyperplane arrangements , author=. Geometric combinatorics , volume=

work page
[39]

, title =

Stanley, Richard P. , title =. 2011 , isbn =

work page 2011
[40]

Cubo Mat

Oxley, James , TITLE =. Cubo Mat. Educ. , FJOURNAL =. 2003 , NUMBER =

work page 2003
[41]

Oxley, James , title =

work page
[42]

Functions of Bounded Variation and Free Discontinuity Problems

Oxley, James , TITLE =. 2011 , PAGES =. doi:10.1093/acprof:oso/9780198566946.001.0001 , URL =

work page doi:10.1093/acprof:oso/9780198566946.001.0001 2011
[43]

2005 , howpublished =

Reiner, Victor , title =. 2005 , howpublished =

work page 2005
[44]

Wilson, R. J. , TITLE =. Amer. Math. Monthly , FJOURNAL =. 1973 , PAGES =. doi:10.2307/2319608 , URL =

work page doi:10.2307/2319608 1973
[45]

, title =

Ardila, Federico and Klivans, Caroline J. , TITLE =. J. Combin. Theory Ser. B , FJOURNAL =. 2006 , NUMBER =. doi:10.1016/j.jctb.2005.06.004 , URL =

work page doi:10.1016/j.jctb.2005.06.004 2006
[46]

Combinatorial analogs of topological zeta functions , volume=

van der Veer, Robin , year=. Combinatorial analogs of topological zeta functions , volume=. Discrete Mathematics , publisher=. doi:10.1016/j.disc.2019.05.035 , number=

work page doi:10.1016/j.disc.2019.05.035 2019
[47]

Kozlov , title =

Eva-Maria Feichtner and Dmitry N. Kozlov , title =. SELECTA MATH. , year =

work page
[48]

De Concini and C

C. De Concini and C. Procesi , title =. Selecta Mathematica , year =

work page
[49]

Denef and F

J. Denef and F. Loeser , journal =. Caracteristiques D'Euler-Poincare, Fonctions Zeta Locales et Modifications Analytiques , volume =

work page
[50]

Katie Gedeon and Nicholas Proudfoot and Benjamin Young , TITLE =. S\'. 2017 , NUMBER =

work page 2017
[51]

Andrews , TITLE =

George E. Andrews , TITLE =

work page