Checking if a polynomial factors into linear factors Let $f\in\mathbb{Q}[x_{1},x_{2},\ldots,x_{n}]$ be a polynomial given
by an arithmetic circuit $C$ of size $s$. Given $C$ as the input,
is there a deterministic algorithm to check whether all the irreducible
factors of $f$ in $\mathbb{Q}[x_{1},x_{2},\ldots,x_{n}]$ are linear
forms? On a related note, given a linear form $l=\sum_{i=1}^{n}l_{i}\cdot x_{i}$,
can we check deterministically whether $l$ is factor of $f$. Of
course, we want running time to be polynomial in both the cases. 
By size, we mean the total bit size. Also, it can be assumed that degree of $f$ is polynomial in $n$.
As far as I know, the best algorithm we have currently to check if $f$ (given by an arithmetic circuit) can be factorized into linear factors is via the randomized algorithm of Kaltofen (PDF) which actually produces blackboxes for all the irreducible factors of $f$, and works over any large enough field. In fact, this problem of polynomial factorization for general circuits was recently shown by Kopparty, Saraf and Shpilka to be equivalent to the problem of blackbox-PIT for general circuits. 

As mentioned by Bruno, if you are interested in checking a the given circuit is divisible by a given $\ell$, then this reduces to a specific PIT problem. We don't know how to do this deterministically in general but I know of one special case where we know how to do this PIT. There is a deterministic poly-time algorithm (PDF) to check if a given $\ell$ divides a given sparse polynomial $f$. 

(Another trivial special case when $f$ is given by a bounded top fan-in depth three circuit. There, $f \bmod \ell$ is also a bounded fan-in depth three circuit and we know how to do PIT in deterministic polynomial time.)