{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:32:57Z","timestamp":1760149977657,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,18]],"date-time":"2023-09-18T00:00:00Z","timestamp":1694995200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Gyeongsang National University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The interplays between topology and algebraic geometry present a set of interesting properties. In this paper, we comprehensively revisit the B\u00e9zout theorem in terms of topology, and we present a topological proof of the theorem considering n-dimensional space. We show the role of topology in understanding the complete and finite intersections of algebraic curves within a topological space. Moreover, we introduce the concept of symmetrically complex translations of roots in a zero-set of a real algebraic curve, which is called a fundamental polynomial, and we show that the resulting complex algebraic curve is additively decomposable into multiple components with varying degrees in a sequence. Interestingly, the symmetrically complex translations of roots in a zero-set of a fundamental polynomial result in the formation of isomorphic topological manifolds if one of the complex translations is kept fixed, and it induces repeated real roots in the fundamental polynomial as a component. A set of numerically simulated examples is included in the paper to illustrate the resulting manifold structures and the associated properties.<\/jats:p>","DOI":"10.3390\/sym15091784","type":"journal-article","created":{"date-parts":[[2023,9,18]],"date-time":"2023-09-18T05:59:06Z","timestamp":1695016746000},"page":"1784","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Topological Approach to the B\u00e9zout\u2019 Theorem and Its Forms"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2667-1446","authenticated-orcid":false,"given":"Susmit","family":"Bagchi","sequence":"first","affiliation":[{"name":"Department of Aerospace and Software Engineering (Informatics), Gyeongsang National University, Jinju 660-701, Republic of Korea"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,18]]},"reference":[{"key":"ref_1","first-page":"459","article-title":"Complexity of Bezout\u2019s theorem. 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