Can I write a Non Homogenerous equation as homogenous

Question

Say I have Fibonacci R.Relation, $$ r^2=r+1 $$ Can I write it as $r^2-r-1=0$?

From what I know a homogeneous equation is an equation equated to zero.

Answer 1

That equation could be made to have homogeneous ("the same") degree by introducing a new variable, say $r^2 - rs - s^2 = 0$.

This sort of thing is typical in studying curves in projective geometry.

Answer 2

In terms of recurrence relations, homogenous relates to linear recurrences. In the Fibonacci example, $F_{n+2}=F_{n+1}+F_n$ is homogeneous since it is linear in the sequence elements without further constants. Scaling the sequence gives another solution. Homogenous linear recurrences with constant coefficients can be solved with an exponential or geometric ansatz, $F_n=r^n$. From that results the quadratic characteristic equation $r^2=r+1$, which is very much non-homogenous.