O'Reilly logo

The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

IV.4   Algebraic Geometry

   János Kollár

1   Introduction

Succinctly put, algebraic geometry is the study of geometry using polynomials and the investigation of polynomials using geometry.

Many of us were taught the beginnings of algebraic geometry in high school, under the name “analytic geometry.” When we say that y = mx + b is the equation of a line L, or that x2 + y2 = r2 describes a circle C of radius r, we establish a basic connection between geometry and algebra.

If we want to find the points where the line L and the circle C intersect, we just substitute mx + b for y in the circle equation to get x2 + (mx + b)2 = r2 and solve the resulting quadratic equation to obtain the x coordinates of the two intersection points.

This simple example ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required