Algebraic variety facts for kids

The twisted cubic is a special kind of algebraic variety.

In mathematics, algebraic varieties are important shapes studied in algebraic geometry. Think of them as the shapes you get when you solve a system of polynomial equations. These equations use numbers like the ones you know (real numbers) or more complex ones.

Imagine you have an equation like y = x + 2. If you draw all the points (x,y) that make this equation true, you get a straight line. That line is a simple algebraic variety! If you have x² + y² = 1, you get a circle. Circles are also algebraic varieties.

Modern math uses a slightly broader idea of what an algebraic variety is. But the main idea is still about shapes made from equations.

Some mathematicians say an "algebraic variety" must be "irreducible." This means you can't break it down into two smaller, separate algebraic varieties. If you can break it down, they might call it an algebraic set instead.

What are Algebraic Varieties?

Algebraic varieties are a bit like manifolds, which are smooth shapes without sharp corners. But varieties can have "singular points." These are like sharp corners or places where the shape crosses itself. For example, a shape like a figure-eight has a singular point where it crosses in the middle.

Around the year 1800, a big idea called the fundamental theorem of algebra connected algebra and geometry. It showed that a polynomial equation (like x² - 4 = 0) has roots (solutions) that are specific points. These points are a geometric idea. This theorem helped show how equations (algebra) are linked to shapes (geometry).

Later, a mathematician named David Hilbert developed something called Hilbert's Nullstellensatz. This is a fancy name for another important link. It connects sets of polynomial equations to specific geometric shapes. This connection is what makes algebraic geometry special. It lets mathematicians solve problems about shapes by using tools from algebra, and vice versa.

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  The affine plane curve y2 = x3 − x. The corresponding projective curve is called an elliptic curve.

See also

In Spanish: Variedad algebraica para niños