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oai:recercat.cat:10459.1/728072024-12-05T21:21:39Zcom_2072_3622col_2072_479130

2024-12-05T21:21:39Z urn:hdl:10459.1/72807 Integrability Conditions for Complex Homogeneous Kukles Systems Giné, Jaume Valls, Claudia Integrability Complex center-focus problem Saddle constants Kukles systems Gröbner Basis In this paper we study the existence of local analytic first integrals for complex polynomial differential systems of the form ẋ = x + Pn(x, y), ẏ = −y, where Pn(x,y) is a homogeneous polynomial of degree n, called the complex homogeneous Kukles systems of degree n. We characterize all the homogeneous Kukles systems of degree n that belong to the Sibirsky ideal. Finally, we provide necessary and sufficient conditions when n = 2,...,7 in order that the complex homogeneous Kukles system has a local analytic first integral computing the saddle constants and using Gröbner bases to find the decomposition of the algebraic variety into its irreducible components. The first author is partially supported by a MINECO/ FEDER grant number 2017-84383-P and an AGAUR (Generalitat de Catalunya) grant number 2017SGR 1276. The second author is partially supported by FCT/Portugal through UID/MAT/04459/2013. 2024-12-05T21:21:39Z 2024-12-05T21:21:39Z 2022-01-24T09:19:52Z 2022-01-24T09:19:52Z 2018 info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion http://hdl.handle.net/10459.1/72807 Reproducció del document publicat a https://doi.org/10.1080/14029251.2018.1494731 Journal of Nonlinear Mathematical Physics, 2018, vol. 25, núm. 3, p. 387-398 cc-by-nc (c) Jaume Giné, Claudia Valls, 2018 info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by-nc/4.0/ Atlantis and Taylor & Francis