{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T12:49:03Z","timestamp":1648644543592},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2010,4,30]],"date-time":"2010-04-30T00:00:00Z","timestamp":1272585600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proceedings of the Edinburgh Mathematical Society"],"published-print":{"date-parts":[[2010,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We characterize the centre variety of the eight-parameter families of real planar polynomial vector fields given, in complex notation, by <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S001309150800014X_inline1\"><jats:alt-text>$\\dot{z}=\\mathrm{i}z+Az^{n_1}\\bar{z}^{j_1}+Bz^{n_2}\\bar{z}^{j_2}+Cz^{n_3}\\bar z^{j_3}+Dz^{n_4}\\bar{z}^{j_4}$<\/jats:alt-text><\/jats:inline-graphic>, where <jats:italic>A,B,C,D<\/jats:italic> \u2208 \u2102 \\{0}, (<jats:italic>n<\/jats:italic><jats:sub>1<\/jats:sub>,<jats:italic>j<\/jats:italic><jats:sub>1<\/jats:sub>) \u2260 (<jats:italic>n<\/jats:italic><jats:sub>2<\/jats:sub>,<jats:italic>j<\/jats:italic><jats:sub>2<\/jats:sub>) \u2260 (<jats:italic>n<\/jats:italic><jats:sub>3<\/jats:sub>,<jats:italic>j<\/jats:italic><jats:sub>3<\/jats:sub>) \u2260 (<jats:italic>n<\/jats:italic><jats:sub>4<\/jats:sub>,<jats:italic>j<\/jats:italic><jats:sub>4<\/jats:sub>), <jats:italic>n<jats:sub>k<\/jats:sub><\/jats:italic> + <jats:italic>j<jats:sub>k<\/jats:sub><\/jats:italic> &gt; 1 for <jats:italic>k<\/jats:italic> = 1,2,3,4, <jats:italic>n<\/jats:italic><jats:sub>1<\/jats:sub> + <jats:italic>j<\/jats:italic><jats:sub>1<\/jats:sub> = <jats:italic>n<\/jats:italic><jats:sub>2<\/jats:sub> + <jats:italic>j<\/jats:italic><jats:sub>2<\/jats:sub> = <jats:italic>n<\/jats:italic><jats:sub>3<\/jats:sub> + <jats:italic>j<\/jats:italic><jats:sub>3<\/jats:sub> = <jats:italic>n<\/jats:italic><jats:sub>4<\/jats:sub> + <jats:italic>j<\/jats:italic><jats:sub>4<\/jats:sub>, |1 \u2212 <jats:italic>n<\/jats:italic><jats:sub>3<\/jats:sub> + <jats:italic>j<\/jats:italic><jats:sub>3<\/jats:sub>| = |1 \u2212 <jats:italic>n<\/jats:italic><jats:sub>2<\/jats:sub> + <jats:italic>j<\/jats:italic><jats:sub>2<\/jats:sub>| \u2260 |1 \u2212 <jats:italic>n<\/jats:italic><jats:sub>1<\/jats:sub> + <jats:italic>j<\/jats:italic><jats:sub>1<\/jats:sub>| and <jats:italic>j<\/jats:italic><jats:sub>4<\/jats:sub> = <jats:italic>n<\/jats:italic><jats:sub>4<\/jats:sub> \u2212 1.<\/jats:p>","DOI":"10.1017\/s001309150800014x","type":"journal-article","created":{"date-parts":[[2010,4,30]],"date-time":"2010-04-30T09:26:57Z","timestamp":1272619617000},"page":"511-530","source":"Crossref","is-referenced-by-count":0,"title":["Centre manifold variety for eight-parameter families of polynomial vector fields of arbitrary degree"],"prefix":"10.1017","volume":"53","author":[{"given":"Cl\u00e0udia","family":"Valls","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2010,4,30]]},"container-title":["Proceedings of the Edinburgh Mathematical Society"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S001309150800014X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,28]],"date-time":"2019-04-28T15:44:27Z","timestamp":1556466267000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S001309150800014X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,4,30]]},"references-count":0,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2010,6]]}},"alternative-id":["S001309150800014X"],"URL":"https:\/\/doi.org\/10.1017\/s001309150800014x","relation":{},"ISSN":["0013-0915","1464-3839"],"issn-type":[{"value":"0013-0915","type":"print"},{"value":"1464-3839","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,4,30]]}}}