Algebraic geometry is a field of mathematics which combines two different branches of study, specifically algebra and linear algebra. Algebra and linear algebra can be defined as (1):

__Algebra:__ A field of mathematics which analyzes polynomial equations using letters and symbols to represent unknown values.

__Linear Algebra:__ The investigation of a system of linear equations using several variables.

Thus, algebraic geometry uses several variables for systems of polynomial equations. Algebraic geometry does not necessarily aim to solve these equations, but rather formulate statements about the geometric structure for the set of solutions belonging to these polynomial equations (algebraic varieties). When studying algebraic geometry, polynomial equations can be of various degrees (denoted as* d*). Polynomial equations of several variables are investigated to discover what their geometric structure might be.

Algebraic geometry is a rather abstract field of mathematics. It is a field which involves a lot of both algebra and geometry, although for a beginner, it may initially seem to be more “algebra heavy”. Algebraic geometry allows for two-dimensional shapes to be defined in three dimensions. For example, if a set of points defined as (x,y,z) satisfy a particular equation, such as x^{2}y^{2} + z^{2} – 3 = 0, this can be translated into a 3-D shape.

In mathematics today, algebraic geometry is often utilized when studying higher complexity math problems related to geometry in other fields. Thus, developing an understanding of algebraic geometry is necessary for multiple fields of mathematics.

Reference:

http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf