  Let $\mathcal{A}$ denote the reduced amalgamated free product of a family
$\mathsf{A}_1, \mathsf{A}_2, ..., \mathsf{A}_n$ of von Neumann algebras over a
von Neumann subalgebra $\Be$ with respect to normal faithful conditional
expectations $\Es_k: \mathsf{A}_k \to \Be$. We investigate the norm in
$L_p(\Al)$ of homogeneous polynomials of a given degree $d$. We first
generalize Voiculescu's inequality to arbitrary degree $d \ge 1$ and indices $1
\le p \le \infty$. This can be regarded as a free analogue of the classical
Rosenthal inequality. Our second result is a length-reduction formula from
which we generalize recent results of Pisier, Ricard and the authors. All
constants in our estimates are independent of $n$ so that we may consider
infinitely many free factors. As applications, we study square functions of
free martingales. More precisely we show that, in contrast with the Khintchine
and Rosenthal inequalities, the free analogue of the Burkholder-Gundy
inequalities does not hold on $L_\infty(\Al)$. At the end of the paper we also
consider Khintchine type inequalities for Shlyakhtenko's generalized circular
systems.
