Ehrhart quasi-polynomials and parallel translations Article Swipe
Akihiro Higashitani
,
Satoshi Murai
,
Masahiko Yoshinaga
·
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2307.08151
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2307.08151
Given a rational polytope $P \subset \mathbb R^d$, the numerical function counting lattice points in the integral dilations of $P$ is known to become a quasi-polynomial, called the Ehrhart quasi-polynomial $\mathrm{ehr}_P$ of $P$. In this paper we study the following problem: Given a rational $d$-polytope $P \subset \mathbb R^d$, is there a nice way to know Ehrhart quasi-polynomials of translated polytopes $P+ \mathbf v$ for all $\mathbf v \in \mathbb Q^d$? We provide a way to compute such Ehrhart quasi-polynomials using a certain toric arrangement and lattice point counting functions of translated cones of $P$. This method allows us to visualize how constituent polynomials of $\mathrm{ehr}_{P+\mathbf v}$ change in the torus $\mathbb R^d/\mathbb Z^d$. We also prove that information of $\mathrm{ehr}_{P+\mathbf v}$ for all $\mathbf v \in \mathbb Q^d$ determines the rational $d$-polytope $P \subset \mathbb R^d$ up to translations by integer vectors, and characterize all rational $d$-polytopes $P \subset \mathbb R^d$ such that $\mathrm{ehr}_{P+\mathbf v}$ is symmetric for all $\mathbf v \in \mathbb Q^d$.
Related Topics
Concepts
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2307.08151
- https://arxiv.org/pdf/2307.08151
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4384816944
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W4384816944Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.48550/arxiv.2307.08151Digital Object Identifier
- Title
-
Ehrhart quasi-polynomials and parallel translationsWork title
- Type
-
preprintOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2023Year of publication
- Publication date
-
2023-07-16Full publication date if available
- Authors
-
Akihiro Higashitani, Satoshi Murai, Masahiko YoshinagaList of authors in order
- Landing page
-
https://arxiv.org/abs/2307.08151Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2307.08151Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://arxiv.org/pdf/2307.08151Direct OA link when available
- Concepts
-
Polytope, Combinatorics, Lattice (music), Rational function, Integer (computer science), Polynomial, Torus, Mathematics, Discrete mathematics, Physics, Pure mathematics, Geometry, Mathematical analysis, Computer science, Acoustics, Programming languageTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
0Total citation count in OpenAlex
- Related works (count)
-
10Other works algorithmically related by OpenAlex
Full payload
| id | https://openalex.org/W4384816944 |
|---|---|
| doi | https://doi.org/10.48550/arxiv.2307.08151 |
| ids.doi | https://doi.org/10.48550/arxiv.2307.08151 |
| ids.openalex | https://openalex.org/W4384816944 |
| fwci | |
| type | preprint |
| title | Ehrhart quasi-polynomials and parallel translations |
| biblio.issue | |
| biblio.volume | |
| biblio.last_page | |
| biblio.first_page | |
| topics[0].id | https://openalex.org/T10948 |
| topics[0].field.id | https://openalex.org/fields/26 |
| topics[0].field.display_name | Mathematics |
| topics[0].score | 0.9998000264167786 |
| topics[0].domain.id | https://openalex.org/domains/3 |
| topics[0].domain.display_name | Physical Sciences |
| topics[0].subfield.id | https://openalex.org/subfields/2607 |
| topics[0].subfield.display_name | Discrete Mathematics and Combinatorics |
| topics[0].display_name | Advanced Combinatorial Mathematics |
| topics[1].id | https://openalex.org/T11428 |
| topics[1].field.id | https://openalex.org/fields/26 |
| topics[1].field.display_name | Mathematics |
| topics[1].score | 0.9941999912261963 |
| topics[1].domain.id | https://openalex.org/domains/3 |
| topics[1].domain.display_name | Physical Sciences |
| topics[1].subfield.id | https://openalex.org/subfields/2602 |
| topics[1].subfield.display_name | Algebra and Number Theory |
| topics[1].display_name | Advanced Mathematical Identities |
| topics[2].id | https://openalex.org/T10588 |
| topics[2].field.id | https://openalex.org/fields/26 |
| topics[2].field.display_name | Mathematics |
| topics[2].score | 0.9782999753952026 |
| topics[2].domain.id | https://openalex.org/domains/3 |
| topics[2].domain.display_name | Physical Sciences |
| topics[2].subfield.id | https://openalex.org/subfields/2610 |
| topics[2].subfield.display_name | Mathematical Physics |
| topics[2].display_name | Mathematical Dynamics and Fractals |
| is_xpac | False |
| apc_list | |
| apc_paid | |
| concepts[0].id | https://openalex.org/C145691206 |
| concepts[0].level | 2 |
| concepts[0].score | 0.9138070344924927 |
| concepts[0].wikidata | https://www.wikidata.org/wiki/Q747980 |
| concepts[0].display_name | Polytope |
| concepts[1].id | https://openalex.org/C114614502 |
| concepts[1].level | 1 |
| concepts[1].score | 0.7708002328872681 |
| concepts[1].wikidata | https://www.wikidata.org/wiki/Q76592 |
| concepts[1].display_name | Combinatorics |
| concepts[2].id | https://openalex.org/C2781204021 |
| concepts[2].level | 2 |
| concepts[2].score | 0.6497394442558289 |
| concepts[2].wikidata | https://www.wikidata.org/wiki/Q6497091 |
| concepts[2].display_name | Lattice (music) |
| concepts[3].id | https://openalex.org/C75190567 |
| concepts[3].level | 2 |
| concepts[3].score | 0.6471767425537109 |
| concepts[3].wikidata | https://www.wikidata.org/wiki/Q41237 |
| concepts[3].display_name | Rational function |
| concepts[4].id | https://openalex.org/C97137487 |
| concepts[4].level | 2 |
| concepts[4].score | 0.5594830513000488 |
| concepts[4].wikidata | https://www.wikidata.org/wiki/Q729138 |
| concepts[4].display_name | Integer (computer science) |
| concepts[5].id | https://openalex.org/C90119067 |
| concepts[5].level | 2 |
| concepts[5].score | 0.5357043147087097 |
| concepts[5].wikidata | https://www.wikidata.org/wiki/Q43260 |
| concepts[5].display_name | Polynomial |
| concepts[6].id | https://openalex.org/C9767117 |
| concepts[6].level | 2 |
| concepts[6].score | 0.471996545791626 |
| concepts[6].wikidata | https://www.wikidata.org/wiki/Q12510 |
| concepts[6].display_name | Torus |
| concepts[7].id | https://openalex.org/C33923547 |
| concepts[7].level | 0 |
| concepts[7].score | 0.42642587423324585 |
| concepts[7].wikidata | https://www.wikidata.org/wiki/Q395 |
| concepts[7].display_name | Mathematics |
| concepts[8].id | https://openalex.org/C118615104 |
| concepts[8].level | 1 |
| concepts[8].score | 0.34632042050361633 |
| concepts[8].wikidata | https://www.wikidata.org/wiki/Q121416 |
| concepts[8].display_name | Discrete mathematics |
| concepts[9].id | https://openalex.org/C121332964 |
| concepts[9].level | 0 |
| concepts[9].score | 0.31905364990234375 |
| concepts[9].wikidata | https://www.wikidata.org/wiki/Q413 |
| concepts[9].display_name | Physics |
| concepts[10].id | https://openalex.org/C202444582 |
| concepts[10].level | 1 |
| concepts[10].score | 0.10675004124641418 |
| concepts[10].wikidata | https://www.wikidata.org/wiki/Q837863 |
| concepts[10].display_name | Pure mathematics |
| concepts[11].id | https://openalex.org/C2524010 |
| concepts[11].level | 1 |
| concepts[11].score | 0.10111141204833984 |
| concepts[11].wikidata | https://www.wikidata.org/wiki/Q8087 |
| concepts[11].display_name | Geometry |
| concepts[12].id | https://openalex.org/C134306372 |
| concepts[12].level | 1 |
| concepts[12].score | 0.07761779427528381 |
| concepts[12].wikidata | https://www.wikidata.org/wiki/Q7754 |
| concepts[12].display_name | Mathematical analysis |
| concepts[13].id | https://openalex.org/C41008148 |
| concepts[13].level | 0 |
| concepts[13].score | 0.07480457425117493 |
| concepts[13].wikidata | https://www.wikidata.org/wiki/Q21198 |
| concepts[13].display_name | Computer science |
| concepts[14].id | https://openalex.org/C24890656 |
| concepts[14].level | 1 |
| concepts[14].score | 0.0 |
| concepts[14].wikidata | https://www.wikidata.org/wiki/Q82811 |
| concepts[14].display_name | Acoustics |
| concepts[15].id | https://openalex.org/C199360897 |
| concepts[15].level | 1 |
| concepts[15].score | 0.0 |
| concepts[15].wikidata | https://www.wikidata.org/wiki/Q9143 |
| concepts[15].display_name | Programming language |
| keywords[0].id | https://openalex.org/keywords/polytope |
| keywords[0].score | 0.9138070344924927 |
| keywords[0].display_name | Polytope |
| keywords[1].id | https://openalex.org/keywords/combinatorics |
| keywords[1].score | 0.7708002328872681 |
| keywords[1].display_name | Combinatorics |
| keywords[2].id | https://openalex.org/keywords/lattice |
| keywords[2].score | 0.6497394442558289 |
| keywords[2].display_name | Lattice (music) |
| keywords[3].id | https://openalex.org/keywords/rational-function |
| keywords[3].score | 0.6471767425537109 |
| keywords[3].display_name | Rational function |
| keywords[4].id | https://openalex.org/keywords/integer |
| keywords[4].score | 0.5594830513000488 |
| keywords[4].display_name | Integer (computer science) |
| keywords[5].id | https://openalex.org/keywords/polynomial |
| keywords[5].score | 0.5357043147087097 |
| keywords[5].display_name | Polynomial |
| keywords[6].id | https://openalex.org/keywords/torus |
| keywords[6].score | 0.471996545791626 |
| keywords[6].display_name | Torus |
| keywords[7].id | https://openalex.org/keywords/mathematics |
| keywords[7].score | 0.42642587423324585 |
| keywords[7].display_name | Mathematics |
| keywords[8].id | https://openalex.org/keywords/discrete-mathematics |
| keywords[8].score | 0.34632042050361633 |
| keywords[8].display_name | Discrete mathematics |
| keywords[9].id | https://openalex.org/keywords/physics |
| keywords[9].score | 0.31905364990234375 |
| keywords[9].display_name | Physics |
| keywords[10].id | https://openalex.org/keywords/pure-mathematics |
| keywords[10].score | 0.10675004124641418 |
| keywords[10].display_name | Pure mathematics |
| keywords[11].id | https://openalex.org/keywords/geometry |
| keywords[11].score | 0.10111141204833984 |
| keywords[11].display_name | Geometry |
| keywords[12].id | https://openalex.org/keywords/mathematical-analysis |
| keywords[12].score | 0.07761779427528381 |
| keywords[12].display_name | Mathematical analysis |
| keywords[13].id | https://openalex.org/keywords/computer-science |
| keywords[13].score | 0.07480457425117493 |
| keywords[13].display_name | Computer science |
| language | en |
| locations[0].id | pmh:oai:arXiv.org:2307.08151 |
| locations[0].is_oa | True |
| locations[0].source.id | https://openalex.org/S4306400194 |
| locations[0].source.issn | |
| locations[0].source.type | repository |
| locations[0].source.is_oa | True |
| locations[0].source.issn_l | |
| locations[0].source.is_core | False |
| locations[0].source.is_in_doaj | False |
| locations[0].source.display_name | arXiv (Cornell University) |
| locations[0].source.host_organization | https://openalex.org/I205783295 |
| locations[0].source.host_organization_name | Cornell University |
| locations[0].source.host_organization_lineage | https://openalex.org/I205783295 |
| locations[0].license | |
| locations[0].pdf_url | https://arxiv.org/pdf/2307.08151 |
| locations[0].version | submittedVersion |
| locations[0].raw_type | |
| locations[0].license_id | |
| locations[0].is_accepted | False |
| locations[0].is_published | False |
| locations[0].raw_source_name | |
| locations[0].landing_page_url | http://arxiv.org/abs/2307.08151 |
| locations[1].id | doi:10.48550/arxiv.2307.08151 |
| locations[1].is_oa | True |
| locations[1].source.id | https://openalex.org/S4306400194 |
| locations[1].source.issn | |
| locations[1].source.type | repository |
| locations[1].source.is_oa | True |
| locations[1].source.issn_l | |
| locations[1].source.is_core | False |
| locations[1].source.is_in_doaj | False |
| locations[1].source.display_name | arXiv (Cornell University) |
| locations[1].source.host_organization | https://openalex.org/I205783295 |
| locations[1].source.host_organization_name | Cornell University |
| locations[1].source.host_organization_lineage | https://openalex.org/I205783295 |
| locations[1].license | cc-by |
| locations[1].pdf_url | |
| locations[1].version | |
| locations[1].raw_type | article |
| locations[1].license_id | https://openalex.org/licenses/cc-by |
| locations[1].is_accepted | False |
| locations[1].is_published | |
| locations[1].raw_source_name | |
| locations[1].landing_page_url | https://doi.org/10.48550/arxiv.2307.08151 |
| indexed_in | arxiv, datacite |
| authorships[0].author.id | https://openalex.org/A5107465100 |
| authorships[0].author.orcid | https://orcid.org/0000-0002-6267-8250 |
| authorships[0].author.display_name | Akihiro Higashitani |
| authorships[0].author_position | first |
| authorships[0].raw_author_name | Higashitani, Akihiro |
| authorships[0].is_corresponding | False |
| authorships[1].author.id | https://openalex.org/A5106407262 |
| authorships[1].author.orcid | |
| authorships[1].author.display_name | Satoshi Murai |
| authorships[1].author_position | middle |
| authorships[1].raw_author_name | Murai, Satoshi |
| authorships[1].is_corresponding | False |
| authorships[2].author.id | https://openalex.org/A5039026089 |
| authorships[2].author.orcid | https://orcid.org/0000-0002-0876-182X |
| authorships[2].author.display_name | Masahiko Yoshinaga |
| authorships[2].author_position | last |
| authorships[2].raw_author_name | Yoshinaga, Masahiko |
| authorships[2].is_corresponding | False |
| has_content.pdf | False |
| has_content.grobid_xml | False |
| is_paratext | False |
| open_access.is_oa | True |
| open_access.oa_url | https://arxiv.org/pdf/2307.08151 |
| open_access.oa_status | green |
| open_access.any_repository_has_fulltext | False |
| created_date | 2025-10-10T00:00:00 |
| display_name | Ehrhart quasi-polynomials and parallel translations |
| has_fulltext | False |
| is_retracted | False |
| updated_date | 2025-11-06T06:51:31.235846 |
| primary_topic.id | https://openalex.org/T10948 |
| primary_topic.field.id | https://openalex.org/fields/26 |
| primary_topic.field.display_name | Mathematics |
| primary_topic.score | 0.9998000264167786 |
| primary_topic.domain.id | https://openalex.org/domains/3 |
| primary_topic.domain.display_name | Physical Sciences |
| primary_topic.subfield.id | https://openalex.org/subfields/2607 |
| primary_topic.subfield.display_name | Discrete Mathematics and Combinatorics |
| primary_topic.display_name | Advanced Combinatorial Mathematics |
| related_works | https://openalex.org/W4280529077, https://openalex.org/W1493684871, https://openalex.org/W2886567582, https://openalex.org/W4390962314, https://openalex.org/W4289703404, https://openalex.org/W2999175164, https://openalex.org/W4387265813, https://openalex.org/W4286224459, https://openalex.org/W2799377959, https://openalex.org/W2083214991 |
| cited_by_count | 0 |
| locations_count | 2 |
| best_oa_location.id | pmh:oai:arXiv.org:2307.08151 |
| best_oa_location.is_oa | True |
| best_oa_location.source.id | https://openalex.org/S4306400194 |
| best_oa_location.source.issn | |
| best_oa_location.source.type | repository |
| best_oa_location.source.is_oa | True |
| best_oa_location.source.issn_l | |
| best_oa_location.source.is_core | False |
| best_oa_location.source.is_in_doaj | False |
| best_oa_location.source.display_name | arXiv (Cornell University) |
| best_oa_location.source.host_organization | https://openalex.org/I205783295 |
| best_oa_location.source.host_organization_name | Cornell University |
| best_oa_location.source.host_organization_lineage | https://openalex.org/I205783295 |
| best_oa_location.license | |
| best_oa_location.pdf_url | https://arxiv.org/pdf/2307.08151 |
| best_oa_location.version | submittedVersion |
| best_oa_location.raw_type | |
| best_oa_location.license_id | |
| best_oa_location.is_accepted | False |
| best_oa_location.is_published | False |
| best_oa_location.raw_source_name | |
| best_oa_location.landing_page_url | http://arxiv.org/abs/2307.08151 |
| primary_location.id | pmh:oai:arXiv.org:2307.08151 |
| primary_location.is_oa | True |
| primary_location.source.id | https://openalex.org/S4306400194 |
| primary_location.source.issn | |
| primary_location.source.type | repository |
| primary_location.source.is_oa | True |
| primary_location.source.issn_l | |
| primary_location.source.is_core | False |
| primary_location.source.is_in_doaj | False |
| primary_location.source.display_name | arXiv (Cornell University) |
| primary_location.source.host_organization | https://openalex.org/I205783295 |
| primary_location.source.host_organization_name | Cornell University |
| primary_location.source.host_organization_lineage | https://openalex.org/I205783295 |
| primary_location.license | |
| primary_location.pdf_url | https://arxiv.org/pdf/2307.08151 |
| primary_location.version | submittedVersion |
| primary_location.raw_type | |
| primary_location.license_id | |
| primary_location.is_accepted | False |
| primary_location.is_published | False |
| primary_location.raw_source_name | |
| primary_location.landing_page_url | http://arxiv.org/abs/2307.08151 |
| publication_date | 2023-07-16 |
| publication_year | 2023 |
| referenced_works_count | 0 |
| abstract_inverted_index.a | 1, 24, 42, 51, 73, 81 |
| abstract_inverted_index.v | 67, 125, 161 |
| abstract_inverted_index.$P | 4, 45, 133, 148 |
| abstract_inverted_index.In | 33 |
| abstract_inverted_index.We | 71, 114 |
| abstract_inverted_index.by | 140 |
| abstract_inverted_index.in | 14, 108 |
| abstract_inverted_index.is | 20, 49, 156 |
| abstract_inverted_index.of | 18, 31, 58, 90, 93, 104, 119 |
| abstract_inverted_index.to | 22, 54, 75, 99, 138 |
| abstract_inverted_index.up | 137 |
| abstract_inverted_index.us | 98 |
| abstract_inverted_index.v$ | 63 |
| abstract_inverted_index.we | 36 |
| abstract_inverted_index.$P$ | 19 |
| abstract_inverted_index.$P+ | 61 |
| abstract_inverted_index.\in | 68, 126, 162 |
| abstract_inverted_index.all | 65, 123, 145, 159 |
| abstract_inverted_index.and | 85, 143 |
| abstract_inverted_index.for | 64, 122, 158 |
| abstract_inverted_index.how | 101 |
| abstract_inverted_index.the | 8, 15, 27, 38, 109, 130 |
| abstract_inverted_index.v}$ | 106, 121, 155 |
| abstract_inverted_index.way | 53, 74 |
| abstract_inverted_index.$P$. | 32, 94 |
| abstract_inverted_index.Q^d$ | 128 |
| abstract_inverted_index.R^d$ | 136, 151 |
| abstract_inverted_index.This | 95 |
| abstract_inverted_index.also | 115 |
| abstract_inverted_index.know | 55 |
| abstract_inverted_index.nice | 52 |
| abstract_inverted_index.such | 77, 152 |
| abstract_inverted_index.that | 117, 153 |
| abstract_inverted_index.this | 34 |
| abstract_inverted_index.Given | 0, 41 |
| abstract_inverted_index.Q^d$. | 164 |
| abstract_inverted_index.Q^d$? | 70 |
| abstract_inverted_index.R^d$, | 7, 48 |
| abstract_inverted_index.Z^d$. | 113 |
| abstract_inverted_index.cones | 92 |
| abstract_inverted_index.known | 21 |
| abstract_inverted_index.paper | 35 |
| abstract_inverted_index.point | 87 |
| abstract_inverted_index.prove | 116 |
| abstract_inverted_index.study | 37 |
| abstract_inverted_index.there | 50 |
| abstract_inverted_index.toric | 83 |
| abstract_inverted_index.torus | 110 |
| abstract_inverted_index.using | 80 |
| abstract_inverted_index.allows | 97 |
| abstract_inverted_index.become | 23 |
| abstract_inverted_index.called | 26 |
| abstract_inverted_index.change | 107 |
| abstract_inverted_index.method | 96 |
| abstract_inverted_index.points | 13 |
| abstract_inverted_index.Ehrhart | 28, 56, 78 |
| abstract_inverted_index.\mathbb | 6, 47, 69, 127, 135, 150, 163 |
| abstract_inverted_index.\mathbf | 62 |
| abstract_inverted_index.\subset | 5, 46, 134, 149 |
| abstract_inverted_index.certain | 82 |
| abstract_inverted_index.compute | 76 |
| abstract_inverted_index.integer | 141 |
| abstract_inverted_index.lattice | 12, 86 |
| abstract_inverted_index.provide | 72 |
| abstract_inverted_index.$\mathbb | 111 |
| abstract_inverted_index.$\mathbf | 66, 124, 160 |
| abstract_inverted_index.counting | 11, 88 |
| abstract_inverted_index.function | 10 |
| abstract_inverted_index.integral | 16 |
| abstract_inverted_index.polytope | 3 |
| abstract_inverted_index.problem: | 40 |
| abstract_inverted_index.rational | 2, 43, 131, 146 |
| abstract_inverted_index.vectors, | 142 |
| abstract_inverted_index.dilations | 17 |
| abstract_inverted_index.following | 39 |
| abstract_inverted_index.functions | 89 |
| abstract_inverted_index.numerical | 9 |
| abstract_inverted_index.polytopes | 60 |
| abstract_inverted_index.symmetric | 157 |
| abstract_inverted_index.visualize | 100 |
| abstract_inverted_index.determines | 129 |
| abstract_inverted_index.translated | 59, 91 |
| abstract_inverted_index.R^d/\mathbb | 112 |
| abstract_inverted_index.arrangement | 84 |
| abstract_inverted_index.constituent | 102 |
| abstract_inverted_index.information | 118 |
| abstract_inverted_index.polynomials | 103 |
| abstract_inverted_index.$d$-polytope | 44, 132 |
| abstract_inverted_index.characterize | 144 |
| abstract_inverted_index.translations | 139 |
| abstract_inverted_index.$d$-polytopes | 147 |
| abstract_inverted_index.$\mathrm{ehr}_P$ | 30 |
| abstract_inverted_index.quasi-polynomial | 29 |
| abstract_inverted_index.quasi-polynomial, | 25 |
| abstract_inverted_index.quasi-polynomials | 57, 79 |
| abstract_inverted_index.$\mathrm{ehr}_{P+\mathbf | 105, 120, 154 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 3 |
| citation_normalized_percentile |