  In this paper, we study how the degrees of the elements in a minimal
$\mu$-basis of a parametrized surface behave. For an arbitrary rational surface
parametrization $P(s,t)=(a_1(s,t),a_2(s,t),a_3(s,t),a_4(s,t)) \in
\mathbb{F}[s,t]^4$ over an infinite field $\mathbb{F}$, we show the existence
of a $\mu$-basis with polynomials bounded in degree by $O(d^{33})$, where
$d=\max(\text{deg}(a_1),\text{deg}(a_2), \text{deg}(a_3), \text{deg}(a_4))$.
Under additional assumptions we can obtain tighter bounds.
