# Generalization of Bezout Identity for Polynomials

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.