A Topological Approach to the Bézout’ Theorem and Its Forms

• Published in: Symmetry (Basel) Vol. 15; no. 9; p. 1784
• Main Author: Bagchi, Susmit
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.09.2023; MDPI
• Subjects: Algebra; Algebraic Geometry; Codes; General Topology; Manifolds (mathematics); Mathematics; Polynomials; Topology; Translations; zero-set; algebraic curve; polynomial; topology
• ISSN: 2073-8994; 2073-8994
• DOI: 10.3390/sym15091784

The interplays between topology and algebraic geometry present a set of interesting properties. In this paper, we comprehensively revisit the Bézout 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.