Prove for $f,g \in \mathbb{Z}[x], \deg f = n, \deg g = m$:
$$|res(f,g)| \leq ||f||_2^m ||g||_2^n \leq (n+1)^{m/2} (m+1)^{n/2} |f||_\infty^m ||g||_\infty^n$$.
Now I have used as definition for the resultant:
$$res(f,g)=lc(f)^{\deg(g)} lc(g)^{\deg(f)} \Pi_{i,j} (\alpha_i - \beta_j)$$
where $\alpha_i,\beta_j$ are the roots of $f,g$ respectively.
It is clear that the leading coefficient of $f$ to the power of $m$ is less or equal that $\sum_{i=1}^n |f_i|^m$ but how do I prove the part with the $2$-norm: $||f||_2^m=(\sum_{i=1}^n |f_i|^2)^{m/2}$
I also don't know to prove the second inequality. Thanks for any help!
EDIT: I have $lc(f) \leq \sum_{i=1}^n |f_i|\leq(\sum_{i=1}^n |f_i|^2)^{1/2}(\sum_{i=1}^n 1)^{1/2}=||f||_2\sqrt{n}$ by Cauchy-Schwartz and then I can raise it to the power of $m$ on both sides. However what do I do with the $\sqrt{n}$ and what do I do with the product of roots in the original definition of the resultant?