The canonical form, scissors congruence and adjoint degrees of polytopes

TL;DR

This paper explores the canonical form as a valuation in scissors congruence, introducing the adjoint polynomial and degree drop as invariants, and characterizes zonotopes through these concepts with new decomposition formulas.

Contribution

It introduces the degree drop invariant, defines the reduced canonical form, and characterizes zonotopes via maximal degree drop, providing new tools for polytope analysis.

Findings

Degree drop measures how much the adjoint polynomial's degree is reduced.

The reduced canonical form is a translation-invariant valuation that vanishes for positive degree drop.

Zonotopes are characterized as polytopes with maximal degree drop.

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.