A Topological Approach to the Bezout' Theorem and Its Forms

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

Automated summary

In this paper, we comprehensively revisit the Bezout theorem from a topological perspective and explore the role of topology in algebraic curve intersections and complex root translations.