A function with the property that multiplying all arguments by a constant
changes the value of the function by a monotonic function of that constant:
F(kV)=g(k)F(V), where F(·) is the homogeneous function, V is a vector of
arguments, k>0 is any constant, and g(·) is some strictly increasing positive function.

Source: Deardorff's Glossary of International Economics

If g(k) = k^{X} then F(·) is said to be homogeneous of degree X.

f(x,y) = 3x^{2} + xy + 2y^{2} is homogeneous of degree 2.

f(x,y,z) = x^{3} + 3x^{2}y + 3xy^{2} + y^{3}
+ 9xyz is homogeneous of degree 3.