0
$\begingroup$

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.

$\endgroup$
1

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.