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SM ISO690:2012 SCHLOMIUK, Dana. Families of vector fields with an algebraic geometric structure. In: Mathematics and IT: Research and Education. 13 iulie 2021, Chişinău. Chișinău, Republica Moldova: 2021, p. 8. 
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Mathematics and IT: Research and Education 2021  
Conferința "Mathematics and IT: Research and Education " Chişinău, Moldova, 13 iulie 2021  


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In 1878 Darboux published a seminal paper on a geometric theory of integrability based on invariant algebraic curves of vector fields. This work was much admired by Poincar´e who published two articles (1891 and 1897) on the subject and proposed a problem still open today “the problem of Poincar´e”. He called this work “admirable” and “masterly” but except for two articles by Painlev´e and Autonne (1891) and one by Dulac (1908) for a long time there were no other developments. Only over a century after the publication of Darboux’ paper, significant new work was published on this topic by Jouanolou (1979) and by Prelle and Singer (1983). The publication of these papers were followed by a flourishing period when new results extended the theory of Darboux turning it into a very active area of research, fully justifying Poincar´e’s enthusiasm. In this lecture I shall present some of the new results obtained from the beginning of this century and up to most recently that point to a sort of a nascent algebraic geometry of polynomial vector fields in which both dynamical and algebraic geometric aspects intertwine. 


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