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arXiv · 2508.04275

The canonical form, scissors congruence and adjoint degrees of polytopes

Tom Baumbach·Ansgar Freyer·Julian Weigert·Martin Winter
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Abstract

We study the canonical form $\Omega$ as a valuation in the context of scissors congruence for polytopes. We identify the degree of its numerator - the adjoint polynomial $\operatorname{adj}_P$ - as an important invariant in this context. More precisely, for a polytope $P$ we define the degree drop that measures how much smaller than expected the degree of the adjoint polynomial of $P$ is. We show that this drop behaves well under various operations, such as decompositions, restrictions to faces, projections, products and Minkowski sums. Next we define the reduced canonical form $\Omega_0$ and show that it is a translation-invariant 1-homogeneous valuation on polytopes that vanishes if and only if $P$ has positive degree drop. Using it we can prove that zonotopes can be characterized as the $d$-polytopes that have maximal possible degree drop $d-1$. We obtain a decomposition formula for $\Omega_0$ that expresses it as a sum of edge-local quantities of $P$. Finally, we discuss valuations $\Omega_s$ that can distinguish higher values of the degree drop.

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Tom Baumbach, Ansgar Freyer, Julian Weigert, Martin Winter. 2025-08-06. The canonical form, scissors congruence and adjoint degrees of polytopes. https://arxiv.org/abs/2508.04275

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