Definition:Algebraic Variety

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

• Definition:Affine Algebraic Variety

• 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