Algebraic geometry

Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations. When there is more than one variable, geometric considerations enter, and are important to understand the phenomenon. One can say that the subject starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations as to find some solution; this does lead into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.

Zeroes of simultaneous polynomials

In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the two-dimensional sphere in three-dimensional Euclidean space mathbb R^3 could be defined as the set of all points (x,y,z) with

Related Topics:
Polynomials - Sphere - Euclidean space

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:x^2+y^2+z^2-1=0.

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A "slanted" circle in mathbb R^3 can be defined as the set of all points (x,y,z) which satisfy the two polynomial equations

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:x^2+y^2+z^2-1=0,

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:x+y+z=0.

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