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Algebraic Equation

An algebraic equation in n variables is an polynomial equation of the form

f(x_1,x_2,...,x_n)=sum_(e_1,...,e_n)c_(e_1,e_2,...,e_n)x_1^(e_1)x_2^(e_2)...x_n^(e_n)=0,

where the coefficients c_(e_1,e_2,...,e_n) are integers (where the exponents e_i are nonnegative integers and the sum is finite).

Examples of algebraic equations are given in the following table.

curveequation
Cayley's sextic4(x^2+y^2-x)^3-27(x^2+y^2)^2=0
eight curvex^4-(x^2-y^2)=0
line through (1,0) and (0,1)x+y-1=0
plane through (1,0,0),x+y+z-1=0
(0,1,0), and (0,0,1)
unit circlex^2+y^2-1=0
unit spherex^2+y^2+z^2-1=0

The roots of an algebraic equation in one variable are known as algebraic numbers.

See also

Algebraic Expression, Algebraic Function, Algebraic Number, Conjugate Elements, Equation, Gröbner Basis, Integer Polynomial, Irreducible Polynomial, Polynomial, Polynomial Roots, Transcendental Equation

Portions of this entry contributed by David Terr

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Cite this as:

Terr, David and Weisstein, Eric W. "Algebraic Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AlgebraicEquation.html

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