Complexity of deciding whether a tropical linear prevariety is a tropical variety
We give an algorithm, with a singly exponential complexity, deciding whether a tropical linear prevariety is a tropical linear variety . The algorithm relies on a criterion to be a tropical linear variety in terms of a duality between the tropical orthogonalization A ⊥ and the double tropical orthog...
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| Veröffentlicht in: | Applicable algebra in engineering, communication and computing communication and computing, 2021-03, Vol.32 (2), p.157-174 |
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| container_title | Applicable algebra in engineering, communication and computing |
| container_volume | 32 |
| creator | Grigoriev, Dima Vorobjov, Nicolai |
| description | We give an algorithm, with a singly exponential complexity, deciding whether a tropical linear
prevariety
is a tropical linear
variety
. The algorithm relies on a criterion to be a tropical linear variety in terms of a duality between the tropical orthogonalization
A
⊥
and the double tropical orthogonalization
A
⊥
⊥
of a subset
A
of the vector space
(
R
∪
{
∞
}
)
n
. We also give an example of a countable family of tropical hyperplanes such that their intersection is not a tropical prevariety. |
| doi_str_mv | 10.1007/s00200-019-00407-w |
| format | Article |
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prevariety
is a tropical linear
variety
. The algorithm relies on a criterion to be a tropical linear variety in terms of a duality between the tropical orthogonalization
A
⊥
and the double tropical orthogonalization
A
⊥
⊥
of a subset
A
of the vector space
(
R
∪
{
∞
}
)
n
. We also give an example of a countable family of tropical hyperplanes such that their intersection is not a tropical prevariety.</description><identifier>ISSN: 0938-1279</identifier><identifier>EISSN: 1432-0622</identifier><identifier>DOI: 10.1007/s00200-019-00407-w</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Algorithms ; Artificial Intelligence ; Complexity ; Computer Hardware ; Computer Science ; Hyperplanes ; Mathematics ; Original Paper ; Symbolic and Algebraic Manipulation ; Theory of Computation</subject><ispartof>Applicable algebra in engineering, communication and computing, 2021-03, Vol.32 (2), p.157-174</ispartof><rights>Springer-Verlag GmbH Germany, part of Springer Nature 2019</rights><rights>Springer-Verlag GmbH Germany, part of Springer Nature 2019.</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c397t-21860ff5b4746cb46e61a52e512bdc8ccc01ce033b7cdddc45be38c9caf5a4e13</citedby><cites>FETCH-LOGICAL-c397t-21860ff5b4746cb46e61a52e512bdc8ccc01ce033b7cdddc45be38c9caf5a4e13</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/s00200-019-00407-w$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00200-019-00407-w$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>230,314,780,784,885,27924,27925,41488,42557,51319</link.rule.ids><backlink>$$Uhttps://hal.science/hal-03047290$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Grigoriev, Dima</creatorcontrib><creatorcontrib>Vorobjov, Nicolai</creatorcontrib><title>Complexity of deciding whether a tropical linear prevariety is a tropical variety</title><title>Applicable algebra in engineering, communication and computing</title><addtitle>AAECC</addtitle><description>We give an algorithm, with a singly exponential complexity, deciding whether a tropical linear
prevariety
is a tropical linear
variety
. The algorithm relies on a criterion to be a tropical linear variety in terms of a duality between the tropical orthogonalization
A
⊥
and the double tropical orthogonalization
A
⊥
⊥
of a subset
A
of the vector space
(
R
∪
{
∞
}
)
n
. We also give an example of a countable family of tropical hyperplanes such that their intersection is not a tropical prevariety.</description><subject>Algorithms</subject><subject>Artificial Intelligence</subject><subject>Complexity</subject><subject>Computer Hardware</subject><subject>Computer Science</subject><subject>Hyperplanes</subject><subject>Mathematics</subject><subject>Original Paper</subject><subject>Symbolic and Algebraic Manipulation</subject><subject>Theory of Computation</subject><issn>0938-1279</issn><issn>1432-0622</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kE1Lw0AQhhdRsFb_gKeAJw-rsx_JJsdS1AoFEfS8bDaTdkuaxN20tf_e1BT15Glg5nlfhoeQawZ3DEDdBwAOQIFlFECCorsTMmJScAoJ56dkBJlIKeMqOycXIawAIMmkGpHXabNuK_x03T5qyqhA6wpXL6LdErsl-shEnW9aZ00VVa5G46PW49Z4h33Ahb_34_aSnJWmCnh1nGPy_vjwNp3R-cvT83Qyp1ZkqqOcpQmUZZxLJRObywQTZmKOMeN5YVNrLTCLIESubFEUVsY5itRm1pSxkcjEmNwOvUtT6da7tfF73RinZ5O5PuxAgFQ8g-2BvRnY1jcfGwydXjUbX_fvaS4z4DwWMu0pPlDWNyF4LH9qGeiDZj1o1r1m_a1Z7_qQGEKhh-sF-t_qf1JfMnKA3A</recordid><startdate>20210301</startdate><enddate>20210301</enddate><creator>Grigoriev, Dima</creator><creator>Vorobjov, Nicolai</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><general>Springer Verlag</general><scope>AAYXX</scope><scope>CITATION</scope><scope>1XC</scope><scope>VOOES</scope></search><sort><creationdate>20210301</creationdate><title>Complexity of deciding whether a tropical linear prevariety is a tropical variety</title><author>Grigoriev, Dima ; Vorobjov, Nicolai</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c397t-21860ff5b4746cb46e61a52e512bdc8ccc01ce033b7cdddc45be38c9caf5a4e13</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Algorithms</topic><topic>Artificial Intelligence</topic><topic>Complexity</topic><topic>Computer Hardware</topic><topic>Computer Science</topic><topic>Hyperplanes</topic><topic>Mathematics</topic><topic>Original Paper</topic><topic>Symbolic and Algebraic Manipulation</topic><topic>Theory of Computation</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Grigoriev, Dima</creatorcontrib><creatorcontrib>Vorobjov, Nicolai</creatorcontrib><collection>CrossRef</collection><collection>Hyper Article en Ligne (HAL)</collection><collection>Hyper Article en Ligne (HAL) (Open Access)</collection><jtitle>Applicable algebra in engineering, communication and computing</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Grigoriev, Dima</au><au>Vorobjov, Nicolai</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Complexity of deciding whether a tropical linear prevariety is a tropical variety</atitle><jtitle>Applicable algebra in engineering, communication and computing</jtitle><stitle>AAECC</stitle><date>2021-03-01</date><risdate>2021</risdate><volume>32</volume><issue>2</issue><spage>157</spage><epage>174</epage><pages>157-174</pages><issn>0938-1279</issn><eissn>1432-0622</eissn><abstract>We give an algorithm, with a singly exponential complexity, deciding whether a tropical linear
prevariety
is a tropical linear
variety
. The algorithm relies on a criterion to be a tropical linear variety in terms of a duality between the tropical orthogonalization
A
⊥
and the double tropical orthogonalization
A
⊥
⊥
of a subset
A
of the vector space
(
R
∪
{
∞
}
)
n
. We also give an example of a countable family of tropical hyperplanes such that their intersection is not a tropical prevariety.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00200-019-00407-w</doi><tpages>18</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0938-1279 |
| ispartof | Applicable algebra in engineering, communication and computing, 2021-03, Vol.32 (2), p.157-174 |
| issn | 0938-1279 1432-0622 |
| language | eng |
| recordid | cdi_hal_primary_oai_HAL_hal_03047290v1 |
| source | SpringerLink Journals |
| subjects | Algorithms Artificial Intelligence Complexity Computer Hardware Computer Science Hyperplanes Mathematics Original Paper Symbolic and Algebraic Manipulation Theory of Computation |
| title | Complexity of deciding whether a tropical linear prevariety is a tropical variety |
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