Homogenous Equations
Homogeneous equations are mathematical expressions in which every term has the same degree. This is a key concept in algebra. It assists in dealing with polynomial equations and differential equations.
In Algebra
In the context of algebra, particularly with polynomials, a homogeneous equation refers to a polynomial equation in which all terms are of the same degree (Washington, L. C., 2008, pp-19-20, Section 2.3):
Polynomial Example: An equation such as Ax^2 + By^2+ Cxy + Dzy + Exz + Fz^2 = 0 (where A, B, C, D, E and F are constants) is homogeneous of degree 2 because each term involves variables, the index of which, when added together, equals 2.
Homogeneous Functions: A function f (x,y,z,…) is called homogeneous of degree n if, for any scalar λ, the function satisfies f (λx,λy,λz,…) = {λ^n}f (x,y,z,…). This property is extensively used in the study of scale-invariant phenomena.
In Differential Equations
Homogeneous equations also appear prominently in the study of differential equations:
Ordinary Differential Equations (ODEs): A homogeneous differential equation (also [1]) is one where the function and its derivatives are proportionally scaled. For instance, the equation y'' + p(x) y' + q(x) y = 0 is a linear homogeneous ODE if p(x) and q(x) are functions of x only.
Partial Differential Equations (PDEs): Similarly, a PDE such as u_{xx} + u_{yy} = 0 (Laplace’s equation) is homogeneous because all terms involve derivatives of the same order, and there are no free terms (non-derivative terms).
Application in Geometry
In projective geometry, homogeneous equations are used to describe geometric figures in a projective space using homogeneous coordinates:
Projective Curves and Surfaces: An equation such as Ax^2 + Bxy + Cy^2 + Dxz + Eyz + Fz^2 = 0 (where A, B, C, D, E and F are constants) is used in projective geometry to define curves and surfaces. Here, x,y,z are not usual Cartesian coordinates but homogeneous coordinates, where the actual location in projective space is represented as a ratio.
Properties and Uses
Homogeneous equations are particularly valued for their symmetry and invariance properties, making them crucial in a wide range of fields. They facilitate the analysis of systems where scaling and proportionality plays a significant role. They are fundamental in situations where absolute sizes are irrelevant, and only ratios matter.
Understanding and solving homogeneous equations involves techniques that exploit their symmetrical properties, such as using substitution methods in differential equations or projective transformations in geometry. This symmetry often allows for simplification in mathematical modelling and problem-solving.
Homogeneous Equations and Elliptic curves
Converting an Elliptic Curve, in the Weierstrass form (in 2 dimensions - x and y) into a Homogenous equation (involving 3 dimensions - x, y, and z), allows that equation to be manipulated, and analysed, in terms of its partial derivatives, in ways that then allow its properties to become clearer than they otherwise would have been (Washington, L. C., 2008, pp-19-20, Section 2.3).
