Minkowski Length of 3D Lattice Polytopes
We study the Minkowski length L ( P ) of a lattice polytope P , which is defined to be the largest number of non-trivial primitive segments whose Minkowski sum lies in P . The Minkowski length represents the largest possible number of factors in a factorization of polynomials with exponent vectors i...
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| Veröffentlicht in: | Discrete & computational geometry 2012-12, Vol.48 (4), p.1137-1158 |
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| container_title | Discrete & computational geometry |
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| creator | Beckwith, Olivia Grimm, Matthew Soprunova, Jenya Weaver, Bradley |
| description | We study the Minkowski length
L
(
P
) of a lattice polytope
P
, which is defined to be the largest number of non-trivial primitive segments whose Minkowski sum lies in
P
. The Minkowski length represents the largest possible number of factors in a factorization of polynomials with exponent vectors in
P
, and shows up in lower bounds for the minimum distance of toric codes. In this paper we give a polytime algorithm for computing
L
(
P
) where
P
is a 3D lattice polytope.
We next study 3D lattice polytopes of Minkowski length 1. In particular, we show that if
Q
, a subpolytope of
P
, is the Minkowski sum of
L
=
L
(
P
) lattice polytopes
Q
i
, each of Minkowski length 1, then the total number of interior lattice points of the polytopes
Q
1
,…,
Q
L
is at most 4. Both results extend previously known results for lattice polygons. Our methods differ substantially from those used in the two-dimensional case. |
| doi_str_mv | 10.1007/s00454-012-9433-5 |
| format | Article |
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L
(
P
) of a lattice polytope
P
, which is defined to be the largest number of non-trivial primitive segments whose Minkowski sum lies in
P
. The Minkowski length represents the largest possible number of factors in a factorization of polynomials with exponent vectors in
P
, and shows up in lower bounds for the minimum distance of toric codes. In this paper we give a polytime algorithm for computing
L
(
P
) where
P
is a 3D lattice polytope.
We next study 3D lattice polytopes of Minkowski length 1. In particular, we show that if
Q
, a subpolytope of
P
, is the Minkowski sum of
L
=
L
(
P
) lattice polytopes
Q
i
, each of Minkowski length 1, then the total number of interior lattice points of the polytopes
Q
1
,…,
Q
L
is at most 4. Both results extend previously known results for lattice polygons. Our methods differ substantially from those used in the two-dimensional case.</description><identifier>ISSN: 0179-5376</identifier><identifier>EISSN: 1432-0444</identifier><identifier>DOI: 10.1007/s00454-012-9433-5</identifier><identifier>CODEN: DCGEER</identifier><language>eng</language><publisher>New York: Springer-Verlag</publisher><subject>Algorithms ; Combinatorics ; Computational Mathematics and Numerical Analysis ; Factorization ; Geometry ; Lattices ; Mathematical analysis ; Mathematics ; Mathematics and Statistics ; Polygons ; Polytopes ; Segments ; Three dimensional ; Vectors (mathematics)</subject><ispartof>Discrete & computational geometry, 2012-12, Vol.48 (4), p.1137-1158</ispartof><rights>Springer Science+Business Media, LLC 2012</rights><rights>Springer Science+Business Media New York 2012</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c344t-acf837ce3536ad221a19a84acafd1ccb06e2108b3b7ee927f04d256dcd504d873</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00454-012-9433-5$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00454-012-9433-5$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Beckwith, Olivia</creatorcontrib><creatorcontrib>Grimm, Matthew</creatorcontrib><creatorcontrib>Soprunova, Jenya</creatorcontrib><creatorcontrib>Weaver, Bradley</creatorcontrib><title>Minkowski Length of 3D Lattice Polytopes</title><title>Discrete & computational geometry</title><addtitle>Discrete Comput Geom</addtitle><description>We study the Minkowski length
L
(
P
) of a lattice polytope
P
, which is defined to be the largest number of non-trivial primitive segments whose Minkowski sum lies in
P
. The Minkowski length represents the largest possible number of factors in a factorization of polynomials with exponent vectors in
P
, and shows up in lower bounds for the minimum distance of toric codes. In this paper we give a polytime algorithm for computing
L
(
P
) where
P
is a 3D lattice polytope.
We next study 3D lattice polytopes of Minkowski length 1. In particular, we show that if
Q
, a subpolytope of
P
, is the Minkowski sum of
L
=
L
(
P
) lattice polytopes
Q
i
, each of Minkowski length 1, then the total number of interior lattice points of the polytopes
Q
1
,…,
Q
L
is at most 4. Both results extend previously known results for lattice polygons. Our methods differ substantially from those used in the two-dimensional case.</description><subject>Algorithms</subject><subject>Combinatorics</subject><subject>Computational Mathematics and Numerical Analysis</subject><subject>Factorization</subject><subject>Geometry</subject><subject>Lattices</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Polygons</subject><subject>Polytopes</subject><subject>Segments</subject><subject>Three dimensional</subject><subject>Vectors (mathematics)</subject><issn>0179-5376</issn><issn>1432-0444</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2012</creationdate><recordtype>article</recordtype><sourceid>8G5</sourceid><sourceid>BENPR</sourceid><sourceid>GUQSH</sourceid><sourceid>M2O</sourceid><recordid>eNp1kD1PwzAQhi0EEqXwA9gisbAY7vwROyMqn1IQDDBbruOUtGlc4lSo_x5XYUBITHfD8766ewg5R7hCAHUdAYQUFJDRQnBO5QGZoOCMghDikEwAVUElV_kxOYlxCQkvQE_I5XPTrcJXXDVZ6bvF8JGFOuO3WWmHoXE-ew3tbggbH0_JUW3b6M9-5pS839-9zR5p-fLwNLspqeNCDNS6WnPlPJc8txVjaLGwWlhn6wqdm0PuGYKe87nyvmCqBlExmVeukmnTik_J5di76cPn1sfBrJvofNvazodtNMhSP2KRvpySiz_oMmz7Ll1nEqAwR63zROFIuT7E2PvabPpmbfudQTB7d2Z0Z5I7s3dnZMqwMRMT2y18_6v539A3_0BvMg</recordid><startdate>20121201</startdate><enddate>20121201</enddate><creator>Beckwith, Olivia</creator><creator>Grimm, Matthew</creator><creator>Soprunova, Jenya</creator><creator>Weaver, Bradley</creator><general>Springer-Verlag</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7SC</scope><scope>7TB</scope><scope>7XB</scope><scope>88I</scope><scope>8AL</scope><scope>8AO</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>8G5</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>FR3</scope><scope>GNUQQ</scope><scope>GUQSH</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K7-</scope><scope>KR7</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M0N</scope><scope>M2O</scope><scope>M2P</scope><scope>M7S</scope><scope>MBDVC</scope><scope>P5Z</scope><scope>P62</scope><scope>PADUT</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>Q9U</scope></search><sort><creationdate>20121201</creationdate><title>Minkowski Length of 3D Lattice Polytopes</title><author>Beckwith, Olivia ; Grimm, Matthew ; Soprunova, Jenya ; Weaver, Bradley</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c344t-acf837ce3536ad221a19a84acafd1ccb06e2108b3b7ee927f04d256dcd504d873</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2012</creationdate><topic>Algorithms</topic><topic>Combinatorics</topic><topic>Computational Mathematics and Numerical Analysis</topic><topic>Factorization</topic><topic>Geometry</topic><topic>Lattices</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Polygons</topic><topic>Polytopes</topic><topic>Segments</topic><topic>Three dimensional</topic><topic>Vectors (mathematics)</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Beckwith, Olivia</creatorcontrib><creatorcontrib>Grimm, Matthew</creatorcontrib><creatorcontrib>Soprunova, Jenya</creatorcontrib><creatorcontrib>Weaver, Bradley</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>Science Database (Alumni Edition)</collection><collection>Computing Database (Alumni Edition)</collection><collection>ProQuest Pharma Collection</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>Research Library (Alumni Edition)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies & Aerospace Collection</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>Engineering Research Database</collection><collection>ProQuest Central Student</collection><collection>Research Library Prep</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>Computer Science Database</collection><collection>Civil Engineering Abstracts</collection><collection>ProQuest Engineering Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Computing Database</collection><collection>Research Library</collection><collection>Science Database</collection><collection>Engineering Database</collection><collection>Research Library (Corporate)</collection><collection>Advanced Technologies & Aerospace Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>Research Library China</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection><collection>ProQuest Central Basic</collection><jtitle>Discrete & computational geometry</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Beckwith, Olivia</au><au>Grimm, Matthew</au><au>Soprunova, Jenya</au><au>Weaver, Bradley</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Minkowski Length of 3D Lattice Polytopes</atitle><jtitle>Discrete & computational geometry</jtitle><stitle>Discrete Comput Geom</stitle><date>2012-12-01</date><risdate>2012</risdate><volume>48</volume><issue>4</issue><spage>1137</spage><epage>1158</epage><pages>1137-1158</pages><issn>0179-5376</issn><eissn>1432-0444</eissn><coden>DCGEER</coden><abstract>We study the Minkowski length
L
(
P
) of a lattice polytope
P
, which is defined to be the largest number of non-trivial primitive segments whose Minkowski sum lies in
P
. The Minkowski length represents the largest possible number of factors in a factorization of polynomials with exponent vectors in
P
, and shows up in lower bounds for the minimum distance of toric codes. In this paper we give a polytime algorithm for computing
L
(
P
) where
P
is a 3D lattice polytope.
We next study 3D lattice polytopes of Minkowski length 1. In particular, we show that if
Q
, a subpolytope of
P
, is the Minkowski sum of
L
=
L
(
P
) lattice polytopes
Q
i
, each of Minkowski length 1, then the total number of interior lattice points of the polytopes
Q
1
,…,
Q
L
is at most 4. Both results extend previously known results for lattice polygons. Our methods differ substantially from those used in the two-dimensional case.</abstract><cop>New York</cop><pub>Springer-Verlag</pub><doi>10.1007/s00454-012-9433-5</doi><tpages>22</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0179-5376 |
| ispartof | Discrete & computational geometry, 2012-12, Vol.48 (4), p.1137-1158 |
| issn | 0179-5376 1432-0444 |
| language | eng |
| recordid | cdi_proquest_miscellaneous_1283711994 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Algorithms Combinatorics Computational Mathematics and Numerical Analysis Factorization Geometry Lattices Mathematical analysis Mathematics Mathematics and Statistics Polygons Polytopes Segments Three dimensional Vectors (mathematics) |
| title | Minkowski Length of 3D Lattice Polytopes |
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