The canonical form, scissors congruence and adjoint degrees of polytopes
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916883826999296 |
|---|---|
| author | Baumbach, Tom Freyer, Ansgar Weigert, Julian Winter, Martin |
| author_facet | Baumbach, Tom Freyer, Ansgar Weigert, Julian Winter, Martin |
| contents | 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 $\operatorname{adj}_P$ - 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 $Ω_0$ 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 $Ω_0$ that expresses it as a sum of edge-local quantities of $P$. Finally, we discuss valuations $Ω_s$ that can distinguish higher values of the degree drop. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_04275 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The canonical form, scissors congruence and adjoint degrees of polytopes Baumbach, Tom Freyer, Ansgar Weigert, Julian Winter, Martin Combinatorics Algebraic Geometry Metric Geometry 51M20, 52B45, 52B12, 14P05 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 $\operatorname{adj}_P$ - 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 $Ω_0$ 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 $Ω_0$ that expresses it as a sum of edge-local quantities of $P$. Finally, we discuss valuations $Ω_s$ that can distinguish higher values of the degree drop. |
| title | The canonical form, scissors congruence and adjoint degrees of polytopes |
| topic | Combinatorics Algebraic Geometry Metric Geometry 51M20, 52B45, 52B12, 14P05 |
| url | https://arxiv.org/abs/2508.04275 |