ALGEBRAIC VARIETY Meaning and
Definition
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An algebraic variety is a fundamental concept in algebraic geometry. It refers to a set of points in a given affine or projective space that is defined by polynomial equations. More precisely, an algebraic variety can be thought of as the zero locus of a set of polynomial equations in n variables.
In the affine case, an algebraic variety is defined as the set of solutions to a system of polynomial equations with coefficients from some fixed field. The set of variables can take values from the field itself. In the projective case, an algebraic variety is defined as the set of solutions to homogeneous polynomial equations. The set of variables takes values from the projective closure of the field.
Algebraic varieties can have different dimensions, which depend on the number of variables involved. For example, a curve in the plane is a one-dimensional variety, while a surface in three-dimensional space is a two-dimensional variety. More generally, an algebraic variety of dimension n is said to be an n-dimensional variety.
The study of algebraic varieties incorporates methods and techniques from algebra, geometry, and topology. It aims to understand the properties, structure, and geometry of the solution sets of polynomial equations. Algebraic varieties have many applications in various branches of mathematics, physics, and computer science.
Etymology of ALGEBRAIC VARIETY
The word "algebraic variety" is composed of two parts:
1. "Algebraic" derives from the Latin word "algebra" which originally came from the Arabic word "al-jabr", meaning "reunion of broken parts". The term "al-jabr" was used by the Persian mathematician Muhammad ibn Musa al-Khwarizmi in the 9th century to refer to the process of solving equations. It was later adopted into Latin as "algebra" in the 12th century, referring to the study of solving equations and manipulating mathematical expressions.
2. "Variety" comes from the Latin word "varietas", meaning "difference" or "diversity". It was adopted into English in the late 16th century to refer to something that exhibits a wide range of different forms or types.
