| Title: | The center problem for homogeneous case of polynomial maps |
|---|
| Authors: | ID Petek, Renato (Author)
ID Ferčec, Brigita (Author)
ID Mencinger, Matej (Author) |
|---|
| Files: |  axioms-15-00370.pdf (275,11 KB)
MD5: 0F1FD3B9742B60C49A00024EA3F14741
 https://www.mdpi.com/2075-1680/15/5/370
|
|---|
| Language: | English |
|---|
| Work type: | Article |
|---|
| Typology: | 1.01 - Original Scientific Article |
|---|
| Organization: | FGPA - Faculty of Civil Engineering, Transportation Engineering and Architecture
FE - Faculty of Energy Technology
FNM - Faculty of Natural Sciences and Mathematics
|
|---|
| Abstract: | 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. |
|---|
| Keywords: | discrete dynamical systems, center variety, polynomial maps, Poincaré return map |
|---|
| Publication status: | Published |
|---|
| Publication version: | Version of Record |
|---|
| Submitted for review: | 01.04.2026 |
|---|
| Article acceptance date: | 13.05.2026 |
|---|
| Publication date: | 15.05.2026 |
|---|
| Publisher: | MDPI |
|---|
| Year of publishing: | 2026 |
|---|
| Number of pages: | 11 str. |
|---|
| Numbering: | Vol. 15, issue 5, [article no.] 370 |
|---|
| PID: | 20.500.12556/DKUM-98229  |
|---|
| UDC: | 517.9 |
|---|
| ISSN on article: | 2075-1680 |
|---|
| COBISS.SI-ID: | 279685635 |
|---|
| DOI: | 10.3390/axioms15050370 |
|---|
| Publication date in DKUM: | 28.05.2026 |
|---|
| Views: | 179 |
|---|
| Downloads: | 10 |
|---|
| Metadata: |    |
|---|
| Categories: | Misc.
|
|---|
|
:
|
Copy citation |
|---|
| | | | Average score: | (0 votes) |
|---|
| Your score: | Voting is allowed only for logged in users. |
|---|
| Share: |  |
|---|
Hover the mouse pointer over a document title to show the abstract or click
on the title to get all document metadata. |
