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Title:           Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three
Subject:         Journal of Dynamics and Differential Equations, https://doi.org/10.1007/s10884-021-10080-8
Keywords:        Poincaré problem,Darboux integrability,Jacobi multiplier,Nonassociative algebras
Author:          Niclas Kruff 
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      Title: Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three
      Destination: Doc-Start
      Children: 
       Item: 
        Title: Abstract
        Destination: abscnt.2
       Item: 
        Title: 1 Introduction
        Destination: section.1
       Item: 
        Title: 2 Preparations
        Destination: section.2
        Children: 
         Item: 
          Title: 2.1 Semi-invariants and Some of Their Properties
          Destination: subsection.2.1
         Item: 
          Title: 2.2 Poincaré Transforms and Stationary Points at Infinity
          Destination: subsection.2.2
       Item: 
        Title: 3 Invariant Hypersurfaces of a Class of Vector Fields
        Destination: section.3
        Children: 
         Item: 
          Title: 3.1 Degree Bounds for Semi-invariants
          Destination: subsection.3.1
         Item: 
          Title: 3.2 Degree Bounds for Jacobi Multipliers
          Destination: subsection.3.2
         Item: 
          Title: 3.3 Reduction of Dimension
          Destination: subsection.3.3
       Item: 
        Title: 4 Dimension Three
        Destination: section.4
        Children: 
         Item: 
          Title: 4.1 Property S and Reduction
          Destination: subsection.4.1
         Item: 
          Title: 4.2 Quadratic Vector Fields in Dimension Three
          Destination: subsection.4.2
       Item: 
        Title: Acknowledgements
        Destination: ackcount.1
       Item: 
        Title: Appendix
        Destination: nsection.1
        Children: 
         Item: 
          Title: Proofs for Sect. 2
          Destination: nsubsection.1.1
         Item: 
          Title: A result of H. Röhrl
          Destination: nsubsection.1.2
         Item: 
          Title: Construction of Quadratic Vector Fields in Dimension Three: Some Details
          Destination: nsubsection.1.3
          Children: 
           Item: 
            Title: The Explicit Expression for System (8)
            Destination: nsubsubsection.1.3.1
           Item: 
            Title: Concerning the Proof of Proposition 4
            Destination: nsubsubsection.1.3.2
           Item: 
            Title: Computing the Seventh Idempotent
            Destination: nsubsubsection.1.3.3
           Item: 
            Title: On the Proofs of Propositions 5 and 6
            Destination: nsubsubsection.1.3.4
           Item: 
            Title: Details for Example 1
            Destination: nsubsubsection.1.3.5
       Item: 
        Title: References
        Destination: myseccnt.1
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    Title: Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three
    Author: Niclas Kruff 
    Subject: Journal of Dynamics and Differential Equations, https://doi.org/10.1007/s10884-021-10080-8
    Keywords: Poincaré problem,Darboux integrability,Jacobi multiplier,Nonassociative algebras
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                  <pdfaSchema:property>
                     <rdf:Seq>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Usual same as prism:doi</pdfaProperty:description>
                           <pdfaProperty:name>DOI</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>The date when a publication was publishe.</pdfaProperty:description>
                           <pdfaProperty:name>MajorVersionDate</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>CrossmarkDomainExclusive</pdfaProperty:description>
                           <pdfaProperty:name>CrossmarkDomainExclusive</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>CrossMarkDomains</pdfaProperty:description>
                           <pdfaProperty:name>CrossMarkDomains</pdfaProperty:name>
                           <pdfaProperty:valueType>seq Text</pdfaProperty:valueType>
                        </rdf:li>
                     </rdf:Seq>
                  </pdfaSchema:property>
               </rdf:li>
               <rdf:li rdf:parseType="Resource">
                  <pdfaSchema:namespaceURI>http://prismstandard.org/namespaces/basic/2.0/</pdfaSchema:namespaceURI>
                  <pdfaSchema:prefix>prism</pdfaSchema:prefix>
                  <pdfaSchema:schema>Prism Schema</pdfaSchema:schema>
                  <pdfaSchema:property>
                     <rdf:Seq>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>This element provides the url for an article or unit of content. The attribute platform is optionally allowed for situations in which multiple URLs must be specified. PRISM recommends that a subset of the PCV platform values, namely “mobile” and “web”, be used in conjunction with this element. NOTE: PRISM recommends against the use of the #other value allowed in the PRISM Platform controlled vocabulary. In lieu of using #other please reach out to the PRISM group at prism-wg@yahoogroups.com to request addition of your term to the Platform Controlled Vocabulary.</pdfaProperty:description>
                           <pdfaProperty:name>url</pdfaProperty:name>
                           <pdfaProperty:valueType>URI</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>The Digital Object Identifier for the article. The DOI may also be used as the dc:identifier. If used as a dc:identifier, the URI form should be captured, and the bare identifier should also be captured using prism:doi. If an alternate unique identifier is used as the required dc:identifier, then the DOI should be specified as a bare identifier within prism:doi only. If the URL associated with a DOI is to be specified, then prism:url may be used in conjunction with prism:doi in order to provide the service endpoint (i.e. the URL). </pdfaProperty:description>
                           <pdfaProperty:name>doi</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>ISSN for an electronic version of the issue in which the resource occurs. Permits publishers to include a second ISSN, identifying an electronic version of the issue in which the resource occurs (therefore e(lectronic)Issn. If used, prism:eIssn MUST contain the ISSN of the electronic version.</pdfaProperty:description>
                           <pdfaProperty:name>issn</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Volume number</pdfaProperty:description>
                           <pdfaProperty:name>volume</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Issue number</pdfaProperty:description>
                           <pdfaProperty:name>number</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Starting page</pdfaProperty:description>
                           <pdfaProperty:name>startingPage</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Ending page</pdfaProperty:description>
                           <pdfaProperty:name>endingPage</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>The aggregation type specifies the unit of aggregation for a content collection. Comment PRISM recommends that the PRISM Aggregation Type Controlled Vocabulary be used to provide values for this element. Note: PRISM recommends against the use of the #other value currently allowed in this controlled vocabulary. In lieu of using #other please reach out to the PRISM group at info@prismstandard.org to request addition of your term to the Aggregation Type Controlled Vocabulary. </pdfaProperty:description>
                           <pdfaProperty:name>aggregationType</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>Title of the magazine, or other publication, in which a resource was/will be published. Typically this will be used to provide the name of the magazine an article appeared in as metadata for the article, along with information such as the article title, the publisher, volume, number, and cover date. Note: Publication name can be used to differentiate between a print magazine and the online version if the names are different such as “magazine” and “magazine.com.”</pdfaProperty:description>
                           <pdfaProperty:name>publicationName</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>Copyright</pdfaProperty:description>
                           <pdfaProperty:name>copyright</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                     </rdf:Seq>
                  </pdfaSchema:property>
               </rdf:li>
               <rdf:li rdf:parseType="Resource">
                  <pdfaSchema:namespaceURI>http://ns.adobe.com/pdf/1.3/</pdfaSchema:namespaceURI>
                  <pdfaSchema:prefix>pdf</pdfaSchema:prefix>
                  <pdfaSchema:schema>Adobe PDF Schema</pdfaSchema:schema>
                  <pdfaSchema:property>
                     <rdf:Seq>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>A name object indicating whether the document has been modified to include trapping information</pdfaProperty:description>
                           <pdfaProperty:name>Trapped</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                     </rdf:Seq>
                  </pdfaSchema:property>
               </rdf:li>
               <rdf:li rdf:parseType="Resource">
                  <pdfaSchema:namespaceURI>http://ns.adobe.com/xap/1.0/mm/</pdfaSchema:namespaceURI>
                  <pdfaSchema:prefix>xmpMM</pdfaSchema:prefix>
                  <pdfaSchema:schema>XMP Media Management Schema</pdfaSchema:schema>
                  <pdfaSchema:property>
                     <rdf:Seq>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>UUID based identifier for specific incarnation of a document</pdfaProperty:description>
                           <pdfaProperty:name>InstanceID</pdfaProperty:name>
                           <pdfaProperty:valueType>URI</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>The common identifier for all versions and renditions of a document.</pdfaProperty:description>
                           <pdfaProperty:name>DocumentID</pdfaProperty:name>
                           <pdfaProperty:valueType>URI</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>The common identifier for all versions and renditions of a document.</pdfaProperty:description>
                           <pdfaProperty:name>OriginalDocumentID</pdfaProperty:name>
                           <pdfaProperty:valueType>URI</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>A reference to the original document from which this one is derived. It is a minimal reference; missing components can be assumed to be unchanged. For example, a new version might only need to specify the instance ID and version number of the previous version, or a rendition might only need to specify the instance ID and rendition class of the original.</pdfaProperty:description>
                           <pdfaProperty:name>DerivedFrom</pdfaProperty:name>
                           <pdfaProperty:valueType>ResourceRef</pdfaProperty:valueType>
                        </rdf:li>
                     </rdf:Seq>
                  </pdfaSchema:property>
                  <pdfaSchema:valueType>
                     <rdf:Seq>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaType:description>Identifies a portion of a document. This can be a position at which the document has been changed since the most recent event history (stEvt:changed). For a resource within an xmpMM:Ingredients list, the ResourceRef uses this type to identify both the portion of the containing document that refers to the resource, and the portion of the referenced resource that is referenced.</pdfaType:description>
                           <pdfaType:namespaceURI>http://ns.adobe.com/xap/1.0/sType/Part#</pdfaType:namespaceURI>
                           <pdfaType:prefix>stPart</pdfaType:prefix>
                           <pdfaType:type>Part</pdfaType:type>
                        </rdf:li>
                     </rdf:Seq>
                  </pdfaSchema:valueType>
               </rdf:li>
               <rdf:li rdf:parseType="Resource">
                  <pdfaSchema:namespaceURI>http://www.aiim.org/pdfa/ns/id/</pdfaSchema:namespaceURI>
                  <pdfaSchema:prefix>pdfaid</pdfaSchema:prefix>
                  <pdfaSchema:schema>PDF/A ID Schema</pdfaSchema:schema>
                  <pdfaSchema:property>
                     <rdf:Seq>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Part of PDF/A standard</pdfaProperty:description>
                           <pdfaProperty:name>part</pdfaProperty:name>
                           <pdfaProperty:valueType>Integer</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Amendment of PDF/A standard</pdfaProperty:description>
                           <pdfaProperty:name>amd</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>Conformance level of PDF/A standard</pdfaProperty:description>
                           <pdfaProperty:name>conformance</pdfaProperty:name>
                           <pdfaProperty:valueType>Text</pdfaProperty:valueType>
                        </rdf:li>
                     </rdf:Seq>
                  </pdfaSchema:property>
               </rdf:li>
               <rdf:li rdf:parseType="Resource">
                  <pdfaSchema:schema>Springer Nature ORCID Schema</pdfaSchema:schema>
                  <pdfaSchema:namespaceURI>http://springernature.com/ns/xmpExtensions/2.0/</pdfaSchema:namespaceURI>
                  <pdfaSchema:prefix>sn</pdfaSchema:prefix>
                  <pdfaSchema:property>
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                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:name>authorInfo</pdfaProperty:name>
                           <pdfaProperty:valueType>Bag AuthorInformation</pdfaProperty:valueType>
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>Author information: contains the name of each author and his/her ORCiD (ORCiD: Open Researcher and Contributor ID). An ORCiD is a persistent identifier (a non-proprietary alphanumeric code) to uniquely identify scientific and other academic authors.</pdfaProperty:description>
                        </rdf:li>
                     </rdf:Seq>
                  </pdfaSchema:property>
                  <pdfaSchema:valueType>
                     <rdf:Seq>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaType:type>AuthorInformation</pdfaType:type>
                           <pdfaType:namespaceURI>http://springernature.com/ns/xmpExtensions/2.0/authorInfo/</pdfaType:namespaceURI>
                           <pdfaType:prefix>author</pdfaType:prefix>
                           <pdfaType:description>Specifies the types of author information: name and ORCID of an author.</pdfaType:description>
                           <pdfaType:field>
                              <rdf:Seq>
                                 <rdf:li rdf:parseType="Resource">
                                    <pdfaField:name>name</pdfaField:name>
                                    <pdfaField:valueType>Text</pdfaField:valueType>
                                    <pdfaField:description>Gives the name of an author.</pdfaField:description>
                                 </rdf:li>
                                 <rdf:li rdf:parseType="Resource">
                                    <pdfaField:name>orcid</pdfaField:name>
                                    <pdfaField:valueType>URI</pdfaField:valueType>
                                    <pdfaField:description>Gives the ORCID of an author.</pdfaField:description>
                                 </rdf:li>
                              </rdf:Seq>
                           </pdfaType:field>
                        </rdf:li>
                     </rdf:Seq>
                  </pdfaSchema:valueType>
               </rdf:li>
               <rdf:li rdf:parseType="Resource">
                  <pdfaSchema:namespaceURI>http://ns.adobe.com/xap/1.0/t/pg/</pdfaSchema:namespaceURI>
                  <pdfaSchema:prefix>xmpTPg</pdfaSchema:prefix>
                  <pdfaSchema:schema>XMP Paged-Text</pdfaSchema:schema>
                  <pdfaSchema:property>
                     <rdf:Seq>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>internal</pdfaProperty:category>
                           <pdfaProperty:description>XMP08 Spec: An ordered array of plate names that are needed to print the document (including any in contained documents).</pdfaProperty:description>
                           <pdfaProperty:name>PlateNames</pdfaProperty:name>
                           <pdfaProperty:valueType>Seq Text</pdfaProperty:valueType>
                        </rdf:li>
                     </rdf:Seq>
                  </pdfaSchema:property>
               </rdf:li>
               <rdf:li rdf:parseType="Resource">
                  <pdfaSchema:namespaceURI>http://www.niso.org/schemas/jav/1.0/</pdfaSchema:namespaceURI>
                  <pdfaSchema:prefix>jav</pdfaSchema:prefix>
                  <pdfaSchema:schema>NISO</pdfaSchema:schema>
                  <pdfaSchema:property>
                     <rdf:Seq>
                        <rdf:li rdf:parseType="Resource">
                           <pdfaProperty:category>external</pdfaProperty:category>
                           <pdfaProperty:description>Values for Journal Article Version are one of the following:&#xA;AO = Author’s Original&#xA;SMUR = Submitted Manuscript Under Review&#xA;AM = Accepted Manuscript&#xA;P = Proof&#xA;VoR = Version of Record&#xA;CVoR = Corrected Version of Record&#xA;EVoR = Enhanced Version of Record</pdfaProperty:description>
                           <pdfaProperty:name>journal_article_version</pdfaProperty:name>
                           <pdfaProperty:valueType>Closed Choice of Text</pdfaProperty:valueType>
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                     </rdf:Seq>
                  </pdfaSchema:property>
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            </rdf:Bag>
         </pdfaExtension:schemas>
      </rdf:Description>
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