Symbolic Computations of First Integrals for Polynomial Vector Fields
In this article, we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral with bounded degree of a polyno...
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| Veröffentlicht in: | Foundations of computational mathematics 2020-08, Vol.20 (4), p.681-752 |
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| container_title | Foundations of computational mathematics |
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| creator | Chèze, Guillaume Combot, Thierry |
| description | In this article, we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral with bounded degree of a polynomial planar vector field. We give probabilistic and deterministic algorithms. The arithmetic complexity of our probabilistic algorithm is in
O
~
(
N
ω
+
1
)
, where
N
is the bound on the degree of a representation of the first integral and
ω
∈
[
2
;
3
]
is the exponent of linear algebra. This result improves previous algorithms. Our algorithms have been implemented in Maple and are available on the authors’ websites. In the last section, we give some examples showing the efficiency of these algorithms. |
| doi_str_mv | 10.1007/s10208-019-09437-9 |
| format | Article |
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O
~
(
N
ω
+
1
)
, where
N
is the bound on the degree of a representation of the first integral and
ω
∈
[
2
;
3
]
is the exponent of linear algebra. This result improves previous algorithms. Our algorithms have been implemented in Maple and are available on the authors’ websites. In the last section, we give some examples showing the efficiency of these algorithms.</description><identifier>ISSN: 1615-3375</identifier><identifier>EISSN: 1615-3383</identifier><identifier>DOI: 10.1007/s10208-019-09437-9</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algebra ; Algorithms ; Applications of Mathematics ; Computer Science ; Economics ; Fields (mathematics) ; Integrals ; Linear algebra ; Linear and Multilinear Algebras ; Math Applications in Computer Science ; Mathematics ; Mathematics and Statistics ; Matrix Theory ; Numerical Analysis ; Polynomials ; Websites</subject><ispartof>Foundations of computational mathematics, 2020-08, Vol.20 (4), p.681-752</ispartof><rights>SFoCM 2019</rights><rights>COPYRIGHT 2020 Springer</rights><rights>SFoCM 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-c536t-ba7dba1db401cb419e5da71c89068fa960b4f0d139bf48960eea0131ffa33f323</citedby><cites>FETCH-LOGICAL-c536t-ba7dba1db401cb419e5da71c89068fa960b4f0d139bf48960eea0131ffa33f323</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/s10208-019-09437-9$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10208-019-09437-9$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>230,314,780,784,885,27922,27923,41486,42555,51317</link.rule.ids><backlink>$$Uhttps://hal.science/hal-03033720$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Chèze, Guillaume</creatorcontrib><creatorcontrib>Combot, Thierry</creatorcontrib><title>Symbolic Computations of First Integrals for Polynomial Vector Fields</title><title>Foundations of computational mathematics</title><addtitle>Found Comput Math</addtitle><description>In this article, we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral with bounded degree of a polynomial planar vector field. We give probabilistic and deterministic algorithms. The arithmetic complexity of our probabilistic algorithm is in
O
~
(
N
ω
+
1
)
, where
N
is the bound on the degree of a representation of the first integral and
ω
∈
[
2
;
3
]
is the exponent of linear algebra. This result improves previous algorithms. Our algorithms have been implemented in Maple and are available on the authors’ websites. In the last section, we give some examples showing the efficiency of these algorithms.</description><subject>Algebra</subject><subject>Algorithms</subject><subject>Applications of Mathematics</subject><subject>Computer Science</subject><subject>Economics</subject><subject>Fields (mathematics)</subject><subject>Integrals</subject><subject>Linear algebra</subject><subject>Linear and Multilinear Algebras</subject><subject>Math Applications in Computer Science</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Matrix Theory</subject><subject>Numerical Analysis</subject><subject>Polynomials</subject><subject>Websites</subject><issn>1615-3375</issn><issn>1615-3383</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNp9kktr4zAUhc3QwqSPPzArQ1dduL3X8kvLEJpJINDSdroVsix5FGwrlZTS_PtR6yElEIoWkg7fuUiHE0W_EG4QoLx1CClUCSBNgGakTOiPaIIF5gkhFTnZn8v8Z3Tm3BoAc4rZJLp72vW16bSIZ6bfbD332gwuNiqea-t8vBy8bC3vXKyMjR9MtxtMr3kXv0jhgzLXsmvcRXSqAiMv_-_n0Z_53fNskazufy9n01UiclL4pOZlU3Ns6gxQ1BlSmTe8RFFRKCrFaQF1pqBBQmuVVeEqJQckqBQnRJGUnEfX49y_vGMbq3tud8xwzRbTFfvQgED4ZApvGNirkd1Y87qVzrO12dohPI-lGQEMc0NSe6rlnWR6UMZbLnrtBJsWBHMsyqwIVHKEauUgQzRmkEoH-YC_OcKH1chei6OG6wNDYLx89y3fOseWT4-HbDqywhrnrFT7JBDYRxvY2AYW2sA-28BoMJHR5AI8tNJ-pfGN6x-YbbNx</recordid><startdate>20200801</startdate><enddate>20200801</enddate><creator>Chèze, Guillaume</creator><creator>Combot, Thierry</creator><general>Springer US</general><general>Springer</general><general>Springer Nature B.V</general><general>Springer Verlag</general><scope>AAYXX</scope><scope>CITATION</scope><scope>ISR</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>1XC</scope></search><sort><creationdate>20200801</creationdate><title>Symbolic Computations of First Integrals for Polynomial Vector Fields</title><author>Chèze, Guillaume ; Combot, Thierry</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c536t-ba7dba1db401cb419e5da71c89068fa960b4f0d139bf48960eea0131ffa33f323</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Algebra</topic><topic>Algorithms</topic><topic>Applications of Mathematics</topic><topic>Computer Science</topic><topic>Economics</topic><topic>Fields (mathematics)</topic><topic>Integrals</topic><topic>Linear algebra</topic><topic>Linear and Multilinear Algebras</topic><topic>Math Applications in Computer Science</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Matrix Theory</topic><topic>Numerical Analysis</topic><topic>Polynomials</topic><topic>Websites</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Chèze, Guillaume</creatorcontrib><creatorcontrib>Combot, Thierry</creatorcontrib><collection>CrossRef</collection><collection>Gale In Context: Science</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</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>Hyper Article en Ligne (HAL)</collection><jtitle>Foundations of computational mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Chèze, Guillaume</au><au>Combot, Thierry</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Symbolic Computations of First Integrals for Polynomial Vector Fields</atitle><jtitle>Foundations of computational mathematics</jtitle><stitle>Found Comput Math</stitle><date>2020-08-01</date><risdate>2020</risdate><volume>20</volume><issue>4</issue><spage>681</spage><epage>752</epage><pages>681-752</pages><issn>1615-3375</issn><eissn>1615-3383</eissn><abstract>In this article, we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral with bounded degree of a polynomial planar vector field. We give probabilistic and deterministic algorithms. The arithmetic complexity of our probabilistic algorithm is in
O
~
(
N
ω
+
1
)
, where
N
is the bound on the degree of a representation of the first integral and
ω
∈
[
2
;
3
]
is the exponent of linear algebra. This result improves previous algorithms. Our algorithms have been implemented in Maple and are available on the authors’ websites. In the last section, we give some examples showing the efficiency of these algorithms.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10208-019-09437-9</doi><tpages>72</tpages><oa>free_for_read</oa></addata></record> |
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| subjects | Algebra Algorithms Applications of Mathematics Computer Science Economics Fields (mathematics) Integrals Linear algebra Linear and Multilinear Algebras Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Polynomials Websites |
| title | Symbolic Computations of First Integrals for Polynomial Vector Fields |
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