For one, we can't take sqrt because then F could send things to C.
You said R^2 to R is significant...I've been playing around with x and y, but every nonlinear function seems to break both sc.mul and addition.
Could you give me an example of a function that satisfies scalar multiplication but not addition?
more specifically, F: R^2 -> R such that F(av)=a F(v) but F(v1 + v2) != F(v1) + F(v2)
The best thing I could come up with is F(x,y)= |x| . This obviously does not satisfy additivity, but...