20.500.12556/DKUM-98229 The center problem for homogeneous case of polynomial maps We study the center problem for polynomial maps y=�(�)=−�−∑�=1∞����+1, arising from homogeneous algebraic curves �+�+∑�=0���−�,���−���=�+�+��(�,�)=0. While explicit conditions were previously known only for low even degrees �=2,4,6,8,10 , their general structure remained conjectural. In this paper we resolve the case �=12 and prove that for all even degrees �=2� , the center condition is completely characterized by two families of algebraic relations: mirror symmetry conditions and alternating-sum conditions. The proof combines algebraic methods with a direct structural argument. In particular, the necessity part is established without relying on explicit formulas for focus quantities, instead, we make use of the involutive property of the associated map and analyze the symmetric difference ��(�,�(�))−��(�(�),�), which leads to a simple and rigorous characterization of the center condition. This provides a complete and conceptually transparent solution of the homogeneous center problem for polynomial maps. discrete dynamical systems center variety polynomial maps Poincaré return map diskretni dinamični sistemi množica centrov polinomske preslikave Poincaréjeva domneva true false true MDPI Angleški jezik Slovenski jezik https://www.mdpi.com/journal/axioms Članek v reviji 2026-05-28 12:31:30 2026-05-28 12:31:31 2026-06-01 08:03:52 0000-00-00 00:00:00 2026 0 0 11 str. issue 5, [article no.] 370 Vol. 15 15 May 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 2026-04-01 2026-05-13 2026-05-15 517.9 279685635 10.3390/axioms15050370 2075-1680 axioms-15-00370.pdf axioms-15-00370.pdf 1 0F1FD3B9742B60C49A00024EA3F14741 f1281bcdd73646ad7d9570e988a45a92388fc047442aa58ce37b056eced24582 a4b53536-5a80-11f1-a150-00155d580e34 https://dk.um.si/Dokument.php?lang=slv&id=206511 https://www.mdpi.com/2075-1680/15/5/370 1 6597772a-5a80-11f1-a150-00155d580e34 https://dk.um.si/Dokument.php?lang=slv&id=206510 Fakulteta za gradbeništvo, prometno inženirstvo in arhitekturo Fakulteta za energetiko Fakulteta za naravoslovje in matematiko 0 0 0