Why is the definition of homogeneous equation $f(tx, ty)= t^n f(x, y)$? My book did define like that. Then I googled a bit & saw Wolfram alpha & The Great Soviet Encyclopedia also wrote the same definition. But no one did explain why the definition is so.
Can anyone please explain what does this definition actually mean? And what is the homogeneity in this definition??
 A: Homogeneity means that f(x,y) has, in a sense, a well-defined degree as a multivariate function. For instance, the bivariate polynomial
\begin{align}
f(x,y) = x^5 y^3 + 3 x^2 y^6 + 7 y^8
\end{align}  
is homogenous of degree 8 because the sum of the degrees of x and y in each term is 8.
Another way to explain it is in terms of consistence of units. For instance, let B be a box with a square base of length x and a height of y. The surface area is given by
\begin{align}
S(x,y) = 2 x^2 + 4 x y.
\end{align}
This is homogenous of degree 2, as it should be. Each terms represents an area, so the sum of the degree of the variables in each term should be 2. So, homogeity formally captures the notion of consistence of units.
A: Homogeneous functions are very important in projective spaces in algebraic geometry.  In projective spaces, there is an equivalence relation where the point $(x_0,\cdots,x_n)$ is related to other points by scaling by $\lambda\not=0$, i.e.,  $(x_0,\cdots,x_n)\sim(\lambda x_0,\cdots,\lambda x_n)$.
In this case, functions on equivalence classes of points become trickier because, in general, for a function $f$, $f(x_0,\cdots,x_n)\not=f(\lambda x_0,\cdots,\lambda x_n)$.  A function where equality holds is a homogeneous function of degree $0$.
More generally, in algebraic geometry, we're interested in zeros of functions on projective spaces.  We can observe that a homogeneous function $g$ (of any degree), $g(x_0,\cdots,x_n)=0$ iff $g(\lambda x_0,\cdots,\lambda x_n)=0$.  Therefore, the question of zeros of homogeneous functions is well-defined for projective spaces.
These functions are called homogeneous because, for polynomials, a homogeneous function is one where the multi-degree of each term is the same see @PatrickJ.Dynes ' answer.  More precisely, for a homogeneous function, each term of the function grows the same way under scaling and the word "homogeneous" refers to the fact that each term grows by the same factor under scaling.
A: Consider the case of two variables. Homogeneous functions are such that if you know the value at a single point $(x_0,y_0$), then you can compute its value at every point $(x,y)=(tx_0,ty_0)$. Therefore, if you know the value of $f$ on all the point of the unit circle, you know its value everywhere, since you can simply "rescale" $f(x_0,y_0)$. Furthermore, the relative growth rate of the function is "homogeneous" in every direction, meaning that, if you double the coordinates, the value of the function is always going to be multiplied by $2^n$.
