Assuming two polynomials $P_1,P_2 \in \mathbb{Z}_p[r]$ of degree $n$, with no common factors, we know that there exist polynomials $Q_1,Q_2$ s.t.: $Q_1P_1 + Q_2P_2 =1$. From Bezout's identity we also know that $deg(Q_i)<n$ for $i=1,2$.

I am wondering how the above is generalized in the case of more than two polynomials. More specifically, given polynomials $P_i$ for $i=1,...,t$ of degree $n$ with $GCD(P_1,...,P_t) = 1$ there exist polynomials $Q_i$ s.t.: $\sum_{i=1}^tQ_iP_i = 1$. What is the maximum degree of these polynomials $Q_i$? Notice that polynomials $P_i$ may have some common factors when taken pairwise, however there is no common factor shared by all $t$ of them,

I can think of examples of where at least some of the $Q_i$'s have degree larger than $n$ but for all the cases I can come up with, the total sum of their degrees is less than $tn$. That is, $\sum_{i=1}^tdeg(Q_i) < tn$, however I am not able to come up with a proof for this claim.

Can someone point out some direction towards such a proof or invalidate it if it false?