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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.03310 |
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| _version_ | 1866909378296152064 |
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| author | Agnarsson, Geir Lawrence, Jim |
| author_facet | Agnarsson, Geir Lawrence, Jim |
| contents | \emph{Minkowski rings} are certain rings of simple functions on
the Euclidean space $W = {\mathbb{R}}^d$
with multiplicative structure derived from Minkowski addition of convex
polytopes. When the ring is (finitely) generated by a set ${\cal{P}}$
of indicator functions of $n$ polytopes then the ring can be presented
as ${\mathbb{C}}[x_1,\ldots,x_n]/I$ when viewed
as a ${\mathbb{C}}$-algebra, where $I$ is the ideal describing all the relations
implied by identities among Minkowski sums of elements of ${\cal{P}}$.
We discuss in detail
the $1$-dimensional case, the $d$-dimensional box case and the affine
Coxeter arrangement in ${\mathbb{R}}^2$ where the convex sets are formed
by closed half-planes with bounding lines making the regular triangular
grid in ${\mathbb{R}}^2$.
We also consider, for a given polytope $P$, the Minkowski ring
$M^\pm_F(P)$ of the collection ${\cal{F}}(P)$
of the nonempty faces of $P$ and their multiplicative inverses.
Finally we prove some general properties of identities
in the Minkowski ring of ${\cal{F}}(P)$; in particular, we show that
Minkowski rings behave well under Cartesian product, namely that
$M^\pm_F(P\times Q)
\cong M^{\pm}_F(P)\otimes M^{\pm}_F(Q)$
as ${\mathbb{C}}$-algebras where $P$ and $Q$ are polytopes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03310 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minkowski ideals and rings Agnarsson, Geir Lawrence, Jim Combinatorics Commutative Algebra 13B25, 13C05, 52B11 \emph{Minkowski rings} are certain rings of simple functions on the Euclidean space $W = {\mathbb{R}}^d$ with multiplicative structure derived from Minkowski addition of convex polytopes. When the ring is (finitely) generated by a set ${\cal{P}}$ of indicator functions of $n$ polytopes then the ring can be presented as ${\mathbb{C}}[x_1,\ldots,x_n]/I$ when viewed as a ${\mathbb{C}}$-algebra, where $I$ is the ideal describing all the relations implied by identities among Minkowski sums of elements of ${\cal{P}}$. We discuss in detail the $1$-dimensional case, the $d$-dimensional box case and the affine Coxeter arrangement in ${\mathbb{R}}^2$ where the convex sets are formed by closed half-planes with bounding lines making the regular triangular grid in ${\mathbb{R}}^2$. We also consider, for a given polytope $P$, the Minkowski ring $M^\pm_F(P)$ of the collection ${\cal{F}}(P)$ of the nonempty faces of $P$ and their multiplicative inverses. Finally we prove some general properties of identities in the Minkowski ring of ${\cal{F}}(P)$; in particular, we show that Minkowski rings behave well under Cartesian product, namely that $M^\pm_F(P\times Q) \cong M^{\pm}_F(P)\otimes M^{\pm}_F(Q)$ as ${\mathbb{C}}$-algebras where $P$ and $Q$ are polytopes. |
| title | Minkowski ideals and rings |
| topic | Combinatorics Commutative Algebra 13B25, 13C05, 52B11 |
| url | https://arxiv.org/abs/2411.03310 |