Polynomial valuations on plane polygons

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.

• The paper establishes that translation invariant simple valuations are uniquely characterized by the area and Hadwiger-Glur invariants, fully resolving the translation scissors congruence problem.
• It systematically develops an algebraic and cohomological framework using cochains, flag functions, and polynomial differential forms to classify valuations on plane polygons.
• The framework informs practical algorithms for polygon equivalence testing and decomposition, with applications in computer graphics and geometric modeling.

Polynomial Valuations on Plane Polygons: An Authoritative Summary

Scissors Congruence and Valuation Theory

The paper provides a comprehensive treatment of the theory of polynomial valuations on plane polygons, with a particular focus on the translation scissors congruence problem. Scissors congruence is defined via the ability to partition one polygon into finitely many pieces and rearrange them with transformations from a group $G$ (e.g., Euclidean isometries, translations) to obtain another polygon. Critical to this framework is the concept of a simple valuation—a real-valued function on polygons that is additive under dissections and invariant under the actions of $G$. The classical Wallace-Bolyai-Gerwien theorem states that area is a complete invariant for Euclidean scissors congruence. However, for translation scissors congruence, the area alone is insufficient; an uncountable family of Hadwiger-Glur valuations, indexed by lines through the origin, is required.

Algebraic Structure of Simple Valuations

The authors systematically develop the algebraic structure of simple valuations, beginning with the linear (1D) case and extending to the plane (2D). In one dimension, every simple valuation on intervals is characterized as a difference of a function at the endpoints, admitting a canonical identification with $Q$-linear functions. Translation invariant simple valuations correspond precisely to functions linear over $Q$, and polynomial simple valuations are tied to polynomial functions on the abelian group $R$.

For plane polygons, valuation theory is elegantly reformulated through cochains and flag functions. Simple valuations are associated with unique normalized cochains on oriented straight line segments, or alternatively as linear combinations of flag valuations—a function on pairs $(a, L)$ where $a$ is a point on a line $L$. The space of all simple valuations is generated by flag valuations subject to explicit relations.

Classification of Translation Invariant Valuations

A central contribution is the explicit classification of translation invariant simple valuations on plane polygons. Every such valuation is a sum of the area and a linear combination of Hadwiger-Glur valuations indexed by lines through the origin:

$$\mu = \xi \circ \text{area} + \sum_L \epsilon_L \circ \text{HGL}_L$$

where $\xi$ and $G$0 are $G$1-linear, and $G$2 is the Hadwiger-Glur valuation relative to line $G$3 with chosen orientation. The uniqueness of this representation is guaranteed under suitable normalization, and it is established (as a formal consequence of linear algebra and cohomological descriptions) that area together with Hadwiger-Glur valuations provide a complete invariant for translation scissors congruence. This resolves the translation scissors congruence problem: two polygons are translation scissors congruent if and only if their area and Hadwiger-Glur invariants coincide for all lines through the origin. Importantly, while there are uncountably many Hadwiger-Glur valuations, only finitely many need to be checked for polygons with finitely many edges.

Cochains, Differential Forms, and Flag Functions

The paper advances further by developing cohomological and polynomial descriptions of valuations. It elucidates that any simple valuation can be represented as an integral of a cochain over the boundary of a polygon. The authors introduce the notion of associated polynomial differential 1-forms and explore exactness conditions along lines for these forms. This enables the decomposition of polynomial valuations into boundary integrals (from degree $G$4 cochains) and interior integrals (from degree $G$5 exact polynomial 1-forms).

Moreover, a valuation is polynomial of degree $G$6 if, for every polygon $G$7, the function $G$8 is a polynomial of degree $G$9. These are characterized in terms of properties of associated cochains and flag functions. The authors rigorously prove that degree $Q$0 polynomial simple valuations correspond to certain polynomial differential 1-forms and polynomial cochains of degree $Q$1, establishing an explicit categorical correspondence in Theorem 10.2.

Applications and Implications

The theoretical apparatus developed yields powerful polynomial dependence results. For example, the value of a polynomial valuation on generalized virtual polygons (parametrized by support numbers) is a polynomial function of these parameters, with degree tightly controlled. This interplay between the combinatorics of polygon cuts and algebraic structures underlies the algebra of virtual polyhedra and Minkowski sums, enriching both the theory of convex geometry and its computational applications.

On a practical front, explicit classification of translation invariant polynomial valuations informs algorithms for polygon equivalence and decomposition, relevant in computer graphics, geometric modeling, and computational algebra. The cochain and differential form perspectives facilitate boundary-interior separation of invariants—advantageous for analytic and symbolic calculations. The algebraic insight into the generators and relations of the valuation space also opens pathways for further exploration in higher dimensions, topological arrangements, and related homology theories.

Contradictory and Strong Results

The paper claims, with complete algebraic justification, that translation invariant valuations are entirely generated by the area and Hadwiger-Glur invariants, with no other independent translation invariant simple valuations existing. This statement is bold, as it rules out the possibility of other geometric invariants playing a role in translation scissors congruence. The careful cohomological argument, the exactness conditions, and the canonical decompositions substantiate this claim.

Future Directions

The structural clarity achieved in this work suggests several avenues for further research:

• Extension of the results to higher dimensions (polyhedra) and more general transformation groups.
• Systematic study of nonsimple valuations (e.g., perimeter) and their reduction to simple valuations.
• Investigation of continuous versus discrete invariants, especially for virtual polygons and polytopes.
• Algorithmic implementation of valuation classification for practical polygon equivalence testing.

In mathematical analysis and geometric combinatorics, the interplay between polynomial functions, cochains, and group actions promises to deepen understanding of equidecomposability and additive measures.

Conclusion

This paper establishes a thorough and rigorous algebraic, cohomological, and polynomial classification of simple and polynomial valuations on plane polygons, culminating in a complete solution to the translation scissors congruence problem. The identification of area and Hadwiger-Glur invariants as fundamental generators offers definitive insight into the structure of translation invariant valuations, with framework extensible to broader contexts in geometric measure and polytope algebra. The methods introduced—cochains, flag functions, and exact polynomial forms—lay solid groundwork for both theoretical advancement and applied algorithms in geometric analysis.