Degree 0 rational function on unit sphere. Suppose I have a degree zero homogenous rational function which has constant value on unit sphere. Does it mean it has constant value over $\mathbb R^n- 0$? If yes, why? By a degree zero homogenous rational function I mean a rational function of the form $P/Q$ where $P$ and $Q$ are homogenous function of the same degree.

Yes it does. If $x$ is in $\mathbb{R}^n - \\{0\\}$, then $$ \frac{P(x)}{Q(x)} = \frac{P(\lambda v)}{Q(\lambda v)}, $$ where $\lambda = \| x \|$ and $v = x/ \|x\|$. Since $P$ and $Q$ are both homogeneous, say of degree $d$, $$ \frac{P(x)}{Q(x)} = \frac{P(\lambda v)}{Q(\lambda v)} =\frac{\lambda^d}{\lambda^d} \frac{P(v)}{Q(v)} = c, $$ where $c$ is the constant that it achieves on the unit sphere (since $v$ is of unit norm, $P(v)/Q(v) = c$).

Therefore, $P/Q$ is constant on all of $\mathbb{R}^n-\\{0\\}$.