| Title: | The center problem for analytic maps with mixed even-degree terms |
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| Authors: | ID Petek, Renato (Author)
ID Ferčec, Brigita (Author)
ID Fernandes, Wilker (Author)
ID Mencinger, Matej (Author) |
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| Files: |  axioms-15-00669.pdf (269,35 KB)
MD5: C7D6AF20670663970F9A86F99E6A08E9
 https://www.mdpi.com/2075-1680/15/9/669
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| Language: | English |
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| Work type: | Article |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | FNM - Faculty of Natural Sciences and Mathematics
FE - Faculty of Energy Technology
FGPA - Faculty of Civil Engineering, Transportation Engineering and Architecture
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| Abstract: | We investigate the center problem for analytic maps implicitly defined by algebraic equations of the form F(x, y) = x + y + ∑ i+k=2αi,kxiyk = 0. Previous studies of homogeneous analytic maps of even degree and of mixed terms of degrees 2 and 4 revealed two recurring algebraic families characterizing the existence of a center at the origin: mirror-symmetry conditions and alternating-sum conditions. In this paper, we resolve the next open case involving mixed terms of degrees 2 and 6 and show that the same algebraic structure extends to mixed terms of degrees 2 and arbitrary even number. Using an approach based on the involutive identity f ( f (x)) = x, we prove that these two families of algebraic conditions completely characterize the existence of a center at the origin. The proof combines algebraic and structural arguments and avoids the use of explicit higher-order focus quantity computations in the general case. |
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| Keywords: | discrete dynamical systems, center variety, analytic maps, Poincaré return map |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Submitted for review: | 28.07.2026 |
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| Article acceptance date: | 24.08.2026 |
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| Publication date: | 07.09.2026 |
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| Publisher: | MDPI |
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| Year of publishing: | 2026 |
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| Number of pages: | 12 str. |
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| Numbering: | Vol. 15, issue 9, [article no.] 669 |
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| PID: | 20.500.12556/DKUM-100500  |
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| UDC: | 517.9 |
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| ISSN on article: | 2075-1680 |
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| COBISS.SI-ID: | 291745027 |
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| DOI: | 10.3390/axioms15090669 |
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| Publication date in DKUM: | 21.09.2026 |
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| Views: | 122 |
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| Downloads: | 3 |
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| Metadata: |    |
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| Categories: | Misc.
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