Evaluating the multilinearization of an arithmetic circuit? Let $p(x_1,\ldots,x_n)$ be a multi-variate polynomial with coefficients over a field $F$. The multilinearization of $p$, denoted by $\hat{p}$, is the result of repeatedly replacing each $x_i^d$ with $d > 1$ by $x_i$. The result is obviously a multilinear polynomial.

Consider the following problem: given an arithmetic circuit $C(x_1,\ldots,x_n)$ over $F$ and given field elements $a_1,\ldots,a_n$, compute $\hat{C}(a_1,\ldots,a_n)$.

Question: Assuming field-arithmetic can be done in unit time, is there a polynomial-time algorithm for this? Added later: I would also be interested in the special case where $C$ is actually a formula (a circuit of fan-out $1$).
In the case that the field $F$ is of size at least $2n$, I think this problem is hard. More specifically, I think that if the above can be efficiently solved for $F$ this large, then CNF-SAT has efficient randomized algorithms. Say we are given a CNF formula $\varphi$. One can easily come up with an arithmetic circuit $C$ that computes an ``arithmetization'' $p$ of $\varphi$, where the polynomial $p$ agrees with the formula $\varphi$ on $0$-$1$ inputs. Consider the multilinearization $q$ of $p$. Note that $q$ agrees with $p$ and hence $\varphi$ on $\{0,1\}^n$.

I claim that $q$ is non-zero iff $\varphi$ is satisfiable. Clearly, if $q=0$, then $\varphi$ cannot be satisfied. For the converse, one can show that any non-zero multilinear polynomial cannot vanish on all of $\{0,1\}^n$. This implies that a non-zero $q$ (and hence the corresponding $\varphi$) does not vanish at some input in $\{0,1\}^n$.

Therefore, checking for satisfiability of $\varphi$ is equivalent to checking if $q$ is non-zero. Say, now, that we could evaluate $q$ over a large field $F$. Then, using the Schwartz-Zippel Lemma, we could identity-test $q$ using an efficient randomized algorithm and check if it is the zero polynomial (the size of $F$ is used to upper bound the error in the Schwartz-Zippel Lemma).