if there exists an integer $N$ such that for any finite extension $E$ of $L$ and for any set of finitely many elements $\alpha_1, \cdots , \alpha_n \in
{{}_\ell\mathrm{Br}}(E)$, there is a finite extension $E'$ of $E$ of degree at most $N$ such that $\alpha_i \otimes_E E' = 0$ for $1 \leq i \leq n$. Using the result of Krashen (\[cor:krashen2\]), we show that if $L$ is a field of characteristic not equal to 2 with $H^M(L, \mu_2^{\otimes M}) = 0$ for some $M \geq 1$ and with the Brauer group of $L$ is $2$-uniformly bounded, then $u(L)$ is finite (\[cor:uniformbound\_uinvariant\]). It looks plausible that there are fields $L$ of finite $u$-invariant with $L$ not uniformly 2-bounded.

The main result of the paper is to prove the uniform $p$-boundedness for the Brauer group of any function field $F$ in one variable over a complete discretely valued field $K$ with residue field $\kappa$, where char$(\kappa) = p$, char$(F) \neq p$ and $[\kappa :
\kappa^p]$ is finite. We also prove the uniform $\ell$-boundedness for the Brauer group of any function field in one variable over a complete discretely valued field with residue field $\kappa$ and $\ell \neq $char$(\kappa)$ under the assumption that 