Homogeneous Function Terminology

Question

Just a question about terminology: Why is it called a homogeneous function? What is "homogeneous" about being able to pull out multiplicative constants out of arguments of functions?

Answer 1

Homogeneous in general is used to describe things where the parts of a whole are similar to one another. Think of a homogeneous liquid mixture, it has the same proportion of each liquid no matter where you zoom in.

Same for homogeneous functions. You multiply it by $t$, and the $t$ enters the mix in equal proportions everywhere (ie. to every input of the function evenly).

So the name comes from the sense of "sameness" or equal distribution that $f$ has under scalar multiplication

Answer 2

It is homogeneous because all terms have the same degree. That is, the same number of variable factors. The Ancient Greeks were known only for dealing with homogeneous polynomials (well, with their geometric algebra). Thus, to such a polynomial as $x^2+2x+1$ would be nonsense, since we're adding a geometric square to a segment to a unit -- a mixture of different, heterogeneous terms. This persisted to a point in Renaissance mathematics, until someone (Descartes) considered that one could multiply segments to get segments, and thus with this freedom from thinking about products as rectangles, cuboids, etc., but as segments, it suddenly made sense to add heterogeneous terms.

But of course, it still remains that homogeneous polynomials are easier to deal with, and all polynomials are sums of homogeneous polynomials.

The extension to functions that are not necessarily polynomial comes from the observation that an algebraic characterisation of homogeneity in polynomials is that a scaling of each variable by some constant scales the polynomial by some power of the constant. This algebraic characterisation makes it easier to generalise to functions that may not even be analytic. This is similar to the extension of the notions of oddness and evenness from polynomials to arbitrary functions by noting that the property didn't essentially depend on the oddness or evenness of the terms of the polynomial, but simply because of the fact that multiplication by $-1$ gives the identity after and only after an even number of iterations, so that substituting $-x$ for $x$ and noting how the function behaves under this transformation is a more general characterisation of the properties, which makes it possible to extend them even to nonanalytic, indeed discontinuous functions.