Vertex Posets, Monotone Path Polytopes, and Chow Polynomials

Botong Wang; Leonid Monin; Mateusz Micha{\l}ek

arxiv: 2604.27515 · v1 · submitted 2026-04-30 · 🧮 math.CO · math.AG

Vertex Posets, Monotone Path Polytopes, and Chow Polynomials

Mateusz Micha{\l}ek , Leonid Monin , Botong Wang This is my paper

Pith reviewed 2026-05-07 09:34 UTC · model grok-4.3

classification 🧮 math.CO math.AG

keywords vertex posetsmonotone path polytopesChow polynomialsh-polynomialsacyclic orientationsBiałynicki-Birula partitionsconvex polytopesstratifications

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The pith

A linear functional on a convex polytope induces a vertex poset whose Chow polynomial equals the h-polynomial of the dual monotone path polytope.

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

For a convex polytope P with a linear functional ℓ nonconstant on every edge, the functional produces an acyclic orientation on the 1-skeleton. This orientation creates positive and negative partitions of P into unions of relative interiors of faces, and the paper shows these partitions are stratifications if and only if each other is. When the induced relation on vertices forms a graded poset, the Chow polynomial of that poset is identical to the h-polynomial of a dual monotone path polytope. Readers care because the result equates an invariant from poset combinatorics with one from polytope geometry.

Core claim

Assuming the induced vertex relation admits the structure of a graded poset, the Chow polynomial of the resulting vertex poset agrees with the h-polynomial of a dual monotone path polytope.

What carries the argument

The vertex poset induced by the acyclic orientation of the polytope edges under the linear functional, which carries the argument by equating its Chow polynomial to the h-polynomial of the associated monotone path polytope.

Load-bearing premise

The relation on vertices induced by the acyclic orientation must form a graded poset.

What would settle it

Take any convex polytope with a linear functional nonconstant on edges such that the induced vertex relation is graded, compute the Chow polynomial of the vertex poset and the h-polynomial of the dual monotone path polytope, and check whether the two polynomials are identical; a mismatch disproves the claim.

Figures

Figures reproduced from arXiv: 2604.27515 by Botong Wang, Leonid Monin, Mateusz Micha{\l}ek.

**Figure 1.** Figure 1: Examples of polygons with directed edges. The previous lemma, combined with proposition below give many examples when O´ (and similarly O`) are stratifications. Proposition 2.24. (1) For i “ 1, 2, let Pi Ă R ni be a polytope and ℓi a linear function on R ni that is nonconstant on every edge of Pi. Assume that the induced partition O ´ Pi is a stratification for both i “ 1, 2. Then the partition O ´ P1ˆP2 o… view at source ↗

**Figure 2.** Figure 2: A figure with two subfigures Remark 4.20. Let us define the following classes of n-dimensional polytopes: Stratn “ tP | P satisfies Assumption 2u, SCHn “ tP | P P Stratn and CHpPq is simpleu, CharKern “ tP | P P SCHn and κP is the characteristic kernel of the poset OP u view at source ↗

read the original abstract

Let $P\subset\mathbb R^n$ be a convex polytope and let $\ell$ be a linear functional which is nonconstant on every edge of $P$. The induced acyclic orientation determines positive and negative Bia{\l}ynicki-Birula type partitions of $P$ into unions of relative interiors of faces. Our first result establishes a duality: the positive partition is a stratification if and only if the negative one is a stratification. Our second result connects poset invariants with monotone path polytopes. Assuming the induced vertex relation admits the structure of a graded poset, we prove that the Chow polynomial of the resulting vertex poset agrees with the $h$-polynomial of a (dual) monotone path polytope.

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 gives a clean duality for positive and negative BB partitions being stratifications, plus a link from Chow polynomials of vertex posets to h-polynomials of monotone path polytopes when the poset is graded.

read the letter

The main points are a duality saying the positive BB-type partition stratifies the polytope exactly when the negative one does, and an equality between the Chow polynomial of the induced vertex poset and the h-polynomial of a dual monotone path polytope, provided the vertex relation forms a graded poset. The duality follows directly from the definitions of the partitions coming out of the acyclic orientation and looks like a useful observation that could shorten some arguments elsewhere. The polynomial equality is the fresher connection, tying poset invariants to the geometry of monotone paths in a way that is not just a restatement of known facts. If the proofs in the full text are complete and self-contained, this gives a concrete way to move computations between the two sides in the cases where the assumption holds. The graded poset condition is the clear limitation. The paper states the assumption explicitly but does not supply a general criterion for when an arbitrary polytope and generic linear functional produce a vertex poset with all maximal chains of equal length. That leaves the second result applicable mainly to special cases rather than broadly, which matches the stress-test concern. No circularity appears in the setup, and the arguments seem to rest on standard geometric and combinatorial facts rather than on the target quantities themselves. This work sits in combinatorial algebraic geometry and polytope theory. Readers who already work with BB decompositions, monotone paths, or Chow polynomials of posets will see the most direct value, especially if they need relations between different polynomial invariants. It has enough original content and formal grounding to merit peer review, even with the scope restriction from the graded assumption. I would send it to referees.

Referee Report

1 major / 0 minor

Summary. The paper considers a convex polytope P in R^n together with a linear functional ℓ that is nonconstant on every edge. The induced acyclic orientation on the vertices yields positive and negative Bialynicki-Birula-type partitions of P into unions of relative interiors of faces. The first result asserts that the positive partition is a stratification if and only if the negative partition is a stratification. The second result states that, assuming the induced vertex relation forms a graded poset, the Chow polynomial of this vertex poset equals the h-polynomial of a dual monotone path polytope.

Significance. If the results hold, the work establishes a direct link between poset-theoretic invariants (Chow polynomials) and geometric invariants of polytopes (h-polynomials of monotone path polytopes). The unconditional duality between the positive and negative stratifications is a clean combinatorial statement that stands on its own and may find applications in the study of acyclic orientations and cell decompositions of polytopes.

major comments (1)

[Abstract / Theorem 2] Abstract / statement of the second result: the equality between the Chow polynomial and the h-polynomial is proved only under the assumption that the induced vertex relation is graded. No general criterion, sufficient condition, or verification that this graded property holds for generic ℓ (or for all P) is supplied. Because the orientation can in principle admit maximal chains of unequal combinatorial length, the assumption is load-bearing and restricts the applicability of the theorem; either a proof that the poset is always graded or an explicit characterization of the pairs (P,ℓ) for which it is graded is needed.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting the role of the graded assumption in the second main result. We address the comment point by point below.

read point-by-point responses

Referee: [Abstract / Theorem 2] Abstract / statement of the second result: the equality between the Chow polynomial and the h-polynomial is proved only under the assumption that the induced vertex relation is graded. No general criterion, sufficient condition, or verification that this graded property holds for generic ℓ (or for all P) is supplied. Because the orientation can in principle admit maximal chains of unequal combinatorial length, the assumption is load-bearing and restricts the applicability of the theorem; either a proof that the poset is always graded or an explicit characterization of the pairs (P,ℓ) for which it is graded is needed.

Authors: We agree that the graded hypothesis is essential and load-bearing. The Chow polynomial of a poset is defined only when the poset is graded (i.e., all maximal chains between comparable elements have the same length), so the statement of Theorem 2 cannot be made without this hypothesis. The manuscript does not assert that the vertex poset induced by an arbitrary acyclic orientation is always graded, nor does it supply a general criterion. We will revise the paper to (i) include a concrete low-dimensional example of a polytope P and functional ℓ for which the induced vertex poset fails to be graded, thereby demonstrating that the assumption is necessary, and (ii) add a short discussion of sufficient conditions under which the poset is graded (for instance, when P is a simplex or when the direction of ℓ is generic with respect to the edge directions of a zonotope). A complete characterization of all pairs (P,ℓ) yielding a graded vertex poset lies beyond the scope of the present work. revision: partial

standing simulated objections not resolved

A complete, explicit characterization of the pairs (P,ℓ) for which the induced vertex poset is graded.

Circularity Check

0 steps flagged

No significant circularity; results derive from independent combinatorial and geometric arguments under explicit assumptions.

full rationale

The paper states two main results. The first establishes a duality: the positive Bia lynicki-Birula partition is a stratification iff the negative one is, derived from the acyclic orientation on the polytope P induced by a generic linear functional ℓ. The second result is explicitly conditional on the vertex relation forming a graded poset and equates the Chow polynomial of that poset to the h-polynomial of the dual monotone path polytope. No derivation step reduces by construction to its own inputs, renames a fitted quantity as a prediction, or relies on load-bearing self-citations whose content is unverified. The grading assumption is stated openly rather than derived or smuggled, and the claims rest on standard poset and polytope theory applied to the given setup. This is a normal non-circular finding for a pure-mathematics paper whose central claims remain independent of the target invariants.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard convex polytope theory plus one explicit domain assumption; no free parameters or new entities are introduced.

axioms (1)

domain assumption The induced vertex relation admits the structure of a graded poset
Explicitly required for the Chow-h polynomial equality in the abstract.

pith-pipeline@v0.9.0 · 5426 in / 1178 out tokens · 54292 ms · 2026-05-07T09:34:33.765910+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · 1 internal anchor

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