On System of Parameters, Local Intersection Multiplicity and Bezout's Theorem

Boďa, Eduard; Vogel, Wolfgang

doi:10.2307/2043026

Cited by 6 publications

(2 citation statements)

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“…, x n ] as β(Zr( f ), Zr(g), E), where E is an irreducible component in Zr( f ) ∩ Zr(g). This leads to an alternative representation of the Bézout theorem, which is given as follows [8].…”

Section: Preliminariesmentioning

confidence: 99%

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A Topological Approach to the Bézout’ Theorem and Its Forms

Bagchi

2023

Symmetry

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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.

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Section: Preliminariesmentioning

confidence: 99%

“…The proof of Theorem 2 is detailed in [8] without resorting to the concepts of topology. However, another alternative version of the local Bézout theorem is presented in [9], which considers a set of n-polynomials with nindeterminates.…”

Section: Preliminariesmentioning

confidence: 99%