Algebraic geometry as the names says, is the concept of geometry in algebra. In other words, the relation of algebra with its geometrical representation is known as algebraic geometry. It is basically, the algebra of rings, especially the ring of polynomials, (where a ring is denoted by a non empty set R, of real numbers satisfying a list of properties) and their graphical representation of the zeroes of the polynomials.

The formal definition of Ring is as follows:
A ring is a non empty set R with two binary compositions, denoted by ' + ' and ' * ' respectively, satisfying the following properties:
1.    a + b belongs to R, for all a, b belonging to R.
2.    a + b = b + a, for all a, b belonging to R.
3.    a + (b + c) = (a + b) + c, for all a, b, c belonging to R.
4.    Property of additive identity: There exists an element denoted by 0 belonging to R, such that: a + 0 = a, for all a belonging to R.
Here, 0 is called as an additive identity or zero element in R.
5.    Property of additive inverse: For each a belonging to R, there exist an element b belonging to R such that: a + b = 0,
Here, b is called an additive inverse or negative of a, and is written as b = -a, so    that a + (-a) = 0.
6.    a, b belongs to R, for all a, b belonging to R.
7.    a * (b * c) = (a * b) * c, for all a, b, c, belonging to R.
8.    For all a, b, c belonging to R,

  •     a * (b + c) = a * b + a * c, which is known as the left distributive law.
  •     (b + c) * a = b * a + c * a, which is known as the right distributive law.
If all the above mentioned 8 properties is satisfied by a set of numbers belonging to R, then we say, the set is a ring, which is denoted as [R, +, *].
If the properties of 1 to 5, is satisfied then we say [R, +] is an Abelian group.
A ring [R, +, *] is called a commutative ring if a * b = b * a for all a, b belonging to R.
A ring [R, +, *] is called a field if it’s non zero elements form an Abelian group with respect to the composition ‘*’ i.e. if R is a field, then 'a' belonging to R, where 'a' not equal to zero, implies a-1 belongs to R.
A polynomial ring is denoted as R [x], in one variable over a commutative ring R.Once we know the ring of polynomials we can easily relate it to the 2-D or 3-D or in any number of fields. For example, any linear polynomial in first order will always represent a straight line in a plane, as shown below:

Algebraic Geometry

There are various applications of algebraic geometry as in the filed of integer programming, or in statistics including game theory, geometric modeling, matching graphs, codes for error correction and many more.

The most important application of algebraic geometry is, for studying the concept of geometry we can use algebraic equations rather than making the exact graph of them.

For example, if a polynomial is given as f (a, b) = a – b2, then the zeroes set of the polynomial will be the curve defined as f (a, b) = 0, which will be the parabola as shown graphically below:
Algebraic Geometry Applications
The geometric interpretation of the polynomial a = b2
Hence, we do not have to draw the graph to check if it is a parabola, we can just look at the equation given to us and can interpret that it is an equation of the parabola. This is main application of algebraic geometry.

Algebraic geometry is widely applied by space astronauts and research scientist as they can immediately relate any equation to its basic curve on any plane, by just looking at the algebraic properties of the polynomials.

For example, consider the two equations as, x2 + y2 + z2 = 1, which is a second order degree in 3 D, space and hence, will represent a sphere and x + y + z = 0, is a polynomial equation in first order in 3 D and hence, will represent a ring, then applying the concept of algebraic geometry, a researcher can analyze its picture without actually drawing the graph as shown below:
 Algebraic Geometry Applications
The general forms of curves are geometrically represented as follows:
 Algebraic Geometry Curve
Applications of Algebraic Geometry are based on the two important theorems of algebraic geometry which are:

Bezouts Theorem:
In case of projective planes and homogenization (process of space enlarging with points at infinity), all curves will intersect with each other, with at least a single point of intersection.

Chow’s theorem:
All projective complex planes are basically algebraic, that is differential geometry deals with same figure as in case of algebraic geometry.

Algebraic geometry is widely applied for finding out the solutions of the algebraic and differential topology including differential geometry, especially in case of solving partial differential equations. It also helps in the filed of homological algebra, commutative algebra, abstract groups, lie groups, categories and sheaves etc. Algebraic geometry is also widely applied in the theory of automorphic and modular forms, analytic geometry and theory of numbers and many more fields of pure mathematics. Algebraic geometry uses the concept of geometry (projective), along with the number theory and its analysis and therefore, it has its application in almost every filed of mathematics and science too.

Algebraic geometry which includes the algebra of the rings of polynomials and the graphical representation of the zeroes of those polynomials has the most important application in the field of pure mathematics.
Algebraic Geometry is based on the principle of regular polynomial functions which are defined on some spaces, which is known as their algebraic varieties. Therefore, the basic principle of algebraic geometry can be studied in the filed of pure mathematics, which is basically to find out the sets of solutions of the given systems of algebraic equations. We can define these definitions and principles of algebraic geometry as follows:

Definition 1: Let k be any field and k [a1, a2, ….am] = k [a] is the algebra of the polynomials in m variables then over the field k, then a system of algebraic equations is denoted as {F = 0}, where F belongs to S, where S is a subset of k [a].

Definition 2: If K is a filed extension of k, then a solution set of S in K, is a set (x1, x2,  …., xm) belonging to km such that for all F belonging to S, then  F (x1, x2,..…. , xm) = 0. That is the set of all points in km, which satisfy a system of polynomial equations with coefficient in K, is known as the solution set of S.

Definition 3: Consider any two systems of algebraic equations S and U over K, which are equivalent, that is, solution set of (S: K) = solution set of (U, K). Then an equivalence class is known as an affine algebraic variety over k.
Some common principles of algebraic geometry which is widely used in number theory, can be explained by knowing the difference between reducible and irreducible polynomial.
A polynomial f belonging to F [t1, t2, ... , tn] will be known as reducible if and only if f = g * h where neither g nor h is a constant; otherwise we will say that f is irreducible. This principle of algebraic geometry has an immediate geometric impact for the hyper surfaces. Consider a polynomial f that has been factored into irreducible, then the hyper surface, defined as X = V(f) will be the union of hyper surfaces Xi = V(gi), and this decomposition of X, where,
X = X1 $\cup$ X2 $\cup$...... $\cup$ Xr into the hyper surfaces will be define as irreducible polynomials which is always unique.
 
Algebraic Geometry Graph
As we can see from the above graph that this graph has 3 parts, out of which one part is a surface and the other 2 parts, are the curves.
Algebraic geometry is the concept of examining the geometric figures, which can be expressed in polynomial equations and then can be further represented by graphs, in a given coordinate system. Note that in case of algebraic geometry, we use algebraic equations only.

For example, if we define a circle r by x2 + y2 = r2, then the figure of this polynomial can be analyzed in detail for its symmetries, intercepts of x and y axis, and some other major properties. Hence, the main use of algebraic geometry is to study the in-depth details of any polynomial equation. The basic curves on a plane on R, can be studied in detail by just looking at the algebraic properties of the polynomials.

Using algebraic geometry, we can relate polynomials to the various dimensional planes like the coordinates plane, or in 3-d or in any number of field.
If we have any linear polynomials is the first order, then it can be related to the straight lines, planes, hyper planes or linear subspaces, as required.
If we have polynomial equations of second order, then we can relate it to the classical conic sections in the affine, Euclidean and projective cases (which means on real and complex numbers)

Using algebraic geometry helps to classify the algebraic varieties, their invariants, singularities, intersections, their topology, deformations and moduli spaces, and formations of arithmetic problems in terms of geometry.

Apart from being the central part of the pure mathematics, algebraic geometry is widely used in the field of control theory, theory of algorithms, image recognition and many more.

Using algebraic geometry creates the relation between a single variable polynomial’s root and its multiplicity with the intersection multiplicity of any point of two given curves, which in turn can be geometrically interpreted. Just in cases of many concepts in algebraic geometry, both of the above mentioned concepts require purely algebraic definitions as can be seen in the above section. We know, that the geometry comes before algebraic calculations, as can be studied historically, but once we know the use of the algebraic geometry and it’s algebraic definitions has been established, we can get the comparison of the algebra with the concepts in the geometry.

Using Algebraic Geometry Example:
Consider the two systems of polynomial equations, out of which, one is the parabola y = x2 and the other is a line x = 1, as shown below:
 
The homogenized system (which means enlarging the space at the point of infinity) will give the precise system Z(Y - Z) = 0 and X = Z, as shown below:
 
This system will have two sets of solutions corresponding to the relative projective points which will be [1; 1; 1] and [0; 1; 0]. The point [1; 1; 1] is will be considered as the projective version of the affine point (1; 1), and the point [0; 1; 0] will be the point at infinity which will be contained in a vertical line.