Bounding the degrees of a minimal $\mu$-basis for a rational surface parametrization

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.