Definition:Algebraic Variety

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Definition

An algebraic variety is the solution set of a system of simultaneous polynomial equations:

\(\ds \map {P_1} {x_1, x_2, \ldots, x_n}\) \(=\) \(\ds 0\)
\(\ds \map {P_2} {x_1, x_2, \ldots, x_n}\) \(=\) \(\ds 0\)
\(\ds \) \(\cdots\) \(\ds \)
\(\ds \map {P_k} {x_1, x_2, \ldots, x_n}\) \(=\) \(\ds 0\)


Examples

Circle

Consider the circle described by Equation of Circle in Cartesian Plane:

$(1): \quad {x_1}^2 + {x_2}^2 - r^2 = 0$

whose radius is $r$.

Then the circle is the solution set of $(1)$.


Also see

  • Results about algebraic varieties can be found here.


Sources