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Newton non-degenerate Foliations on Projective Toric Surfaces Molina Samper, Beatriz Foliación Singular foliations Invariant curves Newton polygons Toric surfaces 1201.01 Geometría Algebraica We prove that the isolated invariant branches of a weak toric type generalized curve de fined over a projective toric ambient sur- faces extend to projective algebraic curves. To do it, we pass through the characterization of the weak toric type foliations in terms of "Newton non-degeneracy" conditions, in the classical sense of Kouchnirenko and Oka. Finally, under the strongest hypothesis of being a toric type foliation, we nd that there is a dichotomy: Either it has rational fi rst integral but does not have isolated invariant branches or it has finitely many global invariant curves and all of them are extending isolated invariant branches. Ministerio de Educación, Cultura y Deporte of Spain (FPU14/02653 grant) and by the Ministerio de Economía y Competitividad from Spain, under the Project “Algebra y geometría en sistemas dinámicos y foliaciones singulares.” (Ref.: MTM2016-77642-C2-1-P) 2024-10-15T15:21:12Z 2024-10-15T15:21:12Z 2022 info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion Moscow Mathematical Journal, 2022, vol. 22, n.3, 493--520 1609-4514 https://uvadoc.uva.es/handle/10324/70833 10.17323/1609-4514-2022-22-3-493-520 eng http://www.mathjournals.org/mmj/ Attribution-NonCommercial-NoDerivatives 4.0 Internacional info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Independent University of Moscow https://uvadoc.uva.es/bitstream/10324/70833/3/REVISE~1.PDF.jpg Hispana TEXT http://creativecommons.org/licenses/by-nc-nd/4.0/ UVaDOC. Repositorio Documental de la Universidad de Valladolid https://uvadoc.uva.es/handle/10324/70833