2025/08/06 by Baumbach, Tom, Freyer, Ansgar, Weigert, Julian +1
#14P05 #51M20 #52B12 #52B45 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2508.04275
We study the canonical form Ω as a valuation in the context of scissors congruence for polytopes. We identify the degree of its numerator - the adjoint polynomial adjP - 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 Ω0 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 Ω0 that expresses it as a sum of edge-local quantities of P. Finally, we discuss valuations Ωs that can distinguish higher values of the degree drop.