Assume that there is polytime algorithm that given C(→x)∈F(→x)C(x⃗ )∈F(x⃗ ) and →aa⃗ computed the result of the multi-linearization of CC on →aa⃗ . (w.l.o.g. I will assume that the output →bb⃗ will be a vector of pp-bit binary numbers bibi is kk iff the bi,kbi,k is one.)
Since P⊆P/polyP⊆P/poly, there is a polysize boolean circuit that given the encoding of the arithmetic circuit and the values for the variables computes the multi-linearization of the arithmetic circuit on the inputs. Let call this circuit MM.
Let CC be an arbitrary arithmetic circuit. Fix the variables of the boolean circuit MM which describe the arithmetic circuit, so we have a boolean circuit computing the multi-linearization of CC on given inputs.
We can turn this circuit into an arithmetic circuit over FpFp by noting that xp−1xp−1 is 11 for all values but 00 so first raise all inputs to the power p−1p−1. Replace each f∧gf∧g gate by multiplication f.gf.g, each f∨gf∨g gate by f+g−f.gf+g−f.g and each ¬f¬f gate by 1−f1−f.
By the assumption we made above about the format of the output, we can turn the output from binary to values over FpFp. Take the output for bibi and combine them to get ∑0≤k≤p−1kbi,k∑0≤k≤p−1kbi,k.
We can also convert the input given as values over FpFp to binary form since there are polynomials passing through any finite number of points. E.g. if we are working in mod3mod3, consider the polynomials 2x(x+1)2x(x+1) and 2x(x+2)2x(x+2) which give the first and the second bits of the input x∈F3x∈F3.
Combining these we have an arithmetic circuit over FpFp computing the multi-linearization of CC with size polynomail in the size of CC.