A Topological Approach to the Bézout’ Theorem and Its Forms

Abstract

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.

The Properties of Topological Manifolds of Simplicial Polynomials

“…This formulation has applications in analyzing convex polytopes as topological objects. It is known that the topological properties have close interrelationships with the elements of algebraic geometry [16][17][18][19]. It was shown that the simplicial triangulations of an affine topological space can be established by employing h − polynomials, and the topology of an algebraic curve changes when the changes in its coefficients force it to pass through singularities [17,18,20].…”

“…Note that, in all cases, the degenerated simplicial polynomials generate isomorphic topological manifolds with varying orientations.Symmetry 2024,16, 102 …”