Let $i \in \{1,\ldots, n\}$, $f_i(x)$ be a univariate polynomial, and $g(x) = \mathsf{GCD}(f_1(x), \ldots,f_n(x))$. According to Bezout identity, there exists $a_i(x)$ such that:
$$\sum_{i \in [n]}a_i(x)f_i(x) = g(x)$$
What will be the degree of the Bezout coefficients, $a_i(x)$?
When $n = 2$, $$a_1(x)f_1(x) + a_2(x)f_2(x) = g(x)$$
and $\mathsf{deg}(a_1(x)) < \mathsf{deg}(f_2(x)) - \mathsf{deg}(g(x))$ and $\mathsf{deg}(a_2(x)) < \mathsf{deg}(f_1(x)) - \mathsf{deg}(g(x))$.
I am not sure what will be the degree of the polynomials $a_i$, when $n > 2$. Any reference is appreciated.