Algebraic geometry over the complex numbers

The Hodge conjecture is one of the seven millennium problems, and is framed within differential geometry and algebraic geometry. It was proposed by William Hodge in 1950 and is currently a stimulus for the development of several theories based on geometry, analysis, and mathematical physics. It proposes a natural condition for the existence of complex submanifolds within a complex manifold. Manifolds are the spaces in which geometric objects can be considered. In complex manifolds, the structure of the space is based on complex numbers, instead of the most intuitive structure of geometry, based on real numbers.

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Algebraic Geometry over the Complex Numbers

10.1007/978-1-4614-1809-2 ◽

2012 ◽

Cited By ~ 1

Author(s):

Donu Arapura

Keyword(s):

Algebraic Geometry ◽

Complex Numbers

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THE ANALYSIS OF IMAGES IN N-POINT GRAVITATIONAL LENS BY METHODS OF ALGEBRAIC GEOMETRY

Acta Polytechnica ◽

10.14311/ap.2017.57.0404 ◽

2017 ◽

Vol 57 (6) ◽

pp. 404

Author(s):

Albert Kotvytskiy ◽

Semen Bronza ◽

Vladimir Shablenko

Keyword(s):

Algebraic Geometry ◽

Fundamental Problem ◽

Polynomial System ◽

Gravitational Lens ◽

Complex Numbers ◽

Gravitational Lenses ◽

Algebraic Version ◽

Several Variables ◽

In The Beginning ◽

Bezout Theorem

<p>This paper is devoted to the study of images in <em>N</em>-point gravitational lenses by methods of algebraic geometry. In the beginning, we carefully define images in algebraic terms. Based on the definition, we show that in this model of gravitational lenses (for a point source), the dimensions of the images can be only 0 and 1. We reduce it to the fundamental problem of classical algebraic geometry - the study of solutions of a polynomial system of equations. Further, we use well-known concepts and theorems. We adapt known or prove new assertions. Sometimes, these statements have a fairly general form and can be applied to other problems of algebraic geometry. In this paper: the criterion for irreducibility of polynomials in several variables over the field of complex numbers is effectively used. In this paper, an algebraic version of the Bezout theorem and some other statements are formulated and proved. We have applied the theorems proved by us to study the imaging of dimensions 1 and 0.</p>

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Local factors valued in normal domains

International Journal of Number Theory ◽

10.1142/s1793042116500159 ◽

2016 ◽

Vol 12 (01) ◽

pp. 249-272 ◽

Cited By ~ 2

Author(s):

Kęstutis Česnavičius

Keyword(s):

Algebraic Geometry ◽

Complex Numbers ◽

Local Factors ◽

Discrete Valuation Rings ◽

Valuation Rings ◽

Standard Techniques

We give an exposition of Deligne’s theory of local [Formula: see text]-factors over fields and discrete valuation rings under the assumption that the theory over the complex numbers is known. We then employ standard techniques from algebraic geometry to deduce the theory of local [Formula: see text]-factors over arbitrary normal integral schemes.

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Spherical Linkages Analysis and Synthesis by Special Unitary Matrices for Solution via Numerical Algebraic Geometry

Volume 5A: 41st Mechanisms and Robotics Conference ◽

10.1115/detc2017-68318 ◽

2017 ◽

Author(s):

Saleh M. Almestiri ◽

Andrew P. Murray ◽

David H. Myszka

Keyword(s):

Algebraic Geometry ◽

Complex Numbers ◽

Numerical Algebraic Geometry ◽

Unknown Parameters ◽

Unitary Matrices ◽

Analysis And Design ◽

Analysis And Synthesis ◽

Spherical Linkages ◽

Four Bar Linkage ◽

Systems Of Polynomial Equations

Numerical algebraic geometry is the field that studies the computation and manipulation of the solution sets of systems of polynomial equations. The goal of this paper is to formulate spherical linkages analysis and design problems via a method suited to employ the tools of numerical algebraic geometry. Specifically, equations are developed using special unitary matrices that naturally use complex numbers to express physical and joint parameters in a mechanical system. Unknown parameters expressed as complex numbers readily admit solution by the methods of numerical algebraic geometry. This work illustrates their use by analyzing the spherical four-bar and Watt I linkages. In addition, special unitary matrices are utilized to solve the five orientation synthesis of a spherical four-bar linkage. Moreover, synthesis equations were formulated for the Watt I linkage and implemented for an eight orientation task. Results obtained from this method are validated by comparison to other published work.

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