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If $f(ta,tb) = f(a,b)$ $\forall t \neq 0$, then can we conclude that $f$ is a homogeneous polynomial of degree 1?

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All constant functions satisfy this. – user1337 Nov 16 '13 at 20:12
Except constants, yes! – Ehsan M. Kermani Nov 16 '13 at 20:13
A homogeneous degree $1$ polynomial would satisfy $f(ta, tb)=t^1 f(a, b)$. – Macavity Nov 16 '13 at 20:18

Homogeneous of degree $1$ would be $f(ta,tb) = t^1f(a,b)$. In your case you have that $f(ta,tb) = f(a,b)=t^0f(a,b)$. This would mean homogeneous of degree $0$.

Here are some nice lecture notes and finer details on "functions homogeneous of degree $0$".

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