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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.