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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): algebraic variety
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): algebraic variety