Why is a homogeneous function called homogeneous?

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Why is a homogeneous function called homogeneous?

When I ask this, I don't mean, "Show me how to algebraically manipulate a function whose input has been multiplied by a constant to get the original function multiplied by the same constant."

I mean--why do we use the word "homogeneous"? That word in particular must have been chosen for a reason; what is it meant to communicate in this context?

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2 Answers

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In a homogeneous polynomial, all terms have the same degree.

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The function of one or several variables that satisfies the following condition: when all independent variables of a function are simultaneously multiplied by the same (arbitrary) factor, the value of the function is multiplied by some power of this factor. In algebraic terms, a function f(x, y, …, u) is said to be homogeneous function of degree n if for all values of x, y, …, u and for any λ

f (λx, λy, …, λu) = λnf(x, y, …, u)

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