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arxiv: 2606.19986 · v1 · pith:JIRS2UXPnew · submitted 2026-06-18 · 🧮 math.MG

Polynomial valuations on plane polygons

Askold Khovanskii , Valentina Kiritchenko , Vladlen Timorin This is my paper

Pith reviewed 2026-06-26 15:18 UTC · model grok-4.3

classification 🧮 math.MG
keywords simple valuationspolynomial valuationsplane polygonstranslation invariancescissors congruencevaluations on polytopes
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The pith

All polynomial simple valuations on plane polygons are described by first listing all simple valuations and then characterizing translation invariance.

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

The paper starts from a description of every simple valuation on plane polygons without extra assumptions. It then isolates the additional restrictions that translation invariance imposes on these valuations. The same steps produce an explicit description of all polynomial simple valuations as a direct extension of the translation-invariant case. A reader would care because the resulting classification organizes the functions that appear in scissors-congruence questions involving translations.

Core claim

Starting from a description of all simple valuations on polygons, the effect of translation invariance is characterized, from which a description of all polynomial simple valuations follows as a direct generalization of the translation invariant theory.

What carries the argument

The characterization of the restrictions imposed by translation invariance when applied to the full set of simple valuations on polygons.

Load-bearing premise

A description of all simple valuations on polygons permits a clean characterization of the effect of translation invariance from which the polynomial case follows directly.

What would settle it

An explicit example of a polynomial simple valuation on a plane polygon whose form lies outside the described family would show the claimed classification is incomplete.

Figures

Figures reproduced from arXiv: 2606.19986 by Askold Khovanskii, Valentina Kiritchenko, Vladlen Timorin.

Figure 1
Figure 1. Figure 1: Left: a coordinate trapezoid Ta,b. Note that θ(a, b) is by definition µ(Ta,b). On the other hand, θ(a ′ , b′ ) = −µ(Ta ′ ,b′). Right: a convex polygon P with vertices a0, . . . , a7, where a0 and a5 are “artificial” vertices marked at the intersections of ∂P with L0. For every simple valuation µ, the value µ(P) can be computed as the sum of ±µ(Ti), where i = 0, . . . , 7, and the sign is minus for T0 and T… view at source ↗
Figure 2
Figure 2. Figure 2: The shaded region represents a part of a convex polygon P near its vertex a. The line L is a support line of P containing a. In the figure, Υa,L(P) = +1 since, as L rotates around a towards P (as the arrow shows), it rotates in the positive direction. i.e., pairs of the form (a, L), where L is an affine line in R 2 , and a ∈ L is a point. The flag valuation Υa,L takes value 0 on a convex polygon P unless a… view at source ↗
Figure 3
Figure 3. Figure 3: Left: the parallelogram Π(u, v) spanned by vec￾tors u and v. Right: additivity of ωµ as a function of its first argument: ωµ(u1 + u2, v) = ωµ(u1, v) + ωµ(u2, v). that ωµ is skew symmetric, i.e., ωµ(v, u) = −ωµ(u, v), for the reason just mentioned. Lemma 5.2. The form ωµ is bilinear over Q. Proof. Due to skew symmetry, it suffices to establish that ωµ(u1 + u2, v) = ωµ(u1, v) + ωµ(u2, v), for all triples of … view at source ↗
Figure 4
Figure 4. Figure 4: A scissors congruence between X \Y and X \Z, where Y , Z ⊂ X are sufficiently small scissors congruent polygons. Left arrow: swapping Y ∩ Z with W in X \ Y . Right arrow: swapping (Y \ Z) ∪ W with Z in X. particular, Zylev’s theorem is proved there in its maximal generality. Theorem 7.2 follows from Theorem 7.3 and abstract linear algebra, as explained above. 8. Polynomial valuations and cochains Below, we… view at source ↗
read the original abstract

Scissors congruence problems involving translations have prompted the study of translation invariant simple valuations. We review this classical theory from a naive and consistent viewpoint: starting from a description of all simple valuations on polygons, we characterize the effect of translation invariance. A description of all polynomial simple valuations is obtained as a bi-product of the adopted approach and as a direct generalization of the translation invariant theory; it appears to be new.

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.

Referee Report

0 major / 1 minor

Summary. The manuscript reviews the classical theory of translation invariant simple valuations on plane polygons from a naive viewpoint: it begins with a description of all simple valuations on polygons, characterizes the effect of imposing translation invariance, and obtains as a byproduct a description of all polynomial simple valuations that directly generalizes the translation-invariant case and appears to be new.

Significance. If the derivations hold, the work supplies a consistent constructive framework that unifies the translation-invariant theory with its polynomial extension. This could streamline the treatment of scissors-congruence problems involving translations and offers an explicit generalization whose novelty is stated with appropriate caution.

minor comments (1)
  1. The abstract is high-level; the introduction would benefit from a short outline of the main constructive steps or a concrete low-dimensional example illustrating the passage from simple to translation-invariant to polynomial valuations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract outlines a high-level constructive approach: begin with the (presumably known) space of all simple valuations on polygons, characterize the subspace satisfying translation invariance, and obtain the polynomial case as a direct generalization and byproduct. No equations, fitted parameters, self-citations, or ansatzes are exhibited that reduce any claimed result to its own inputs by construction. The derivation chain is presented as a review from a new viewpoint on established theory rather than a self-referential prediction or uniqueness theorem imported from the authors' prior work. Absent specific load-bearing steps in the provided text that collapse to tautology, the paper's central claims remain independent of the inputs described.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no information on free parameters, background axioms, or new entities introduced in the derivations.

pith-pipeline@v0.9.1-grok · 5582 in / 950 out tokens · 20047 ms · 2026-06-26T15:18:51.048901+00:00 · methodology

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Works this paper leans on

14 extracted references · 1 canonical work pages

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