Consider a binary finite field $F = GF[2^{n}]$ with addition and multiplication denoted by $\oplus$ and $*$, respectively. Let me represent the elements of $F$ by $n$-bit strings, which means that addition is just the bitwise XOR.
Now, suppose that I want to approximate the monomial $X * Y$ as a sum of $f(X) \oplus g(Y)$, where $f, g$ are arbitrary functions from $F$ to itself. There are $2^{n} \cdot 2^{n} = 2^{2n}$ pairs of inputs. How many of them can I be correct on?
Alternatively, if we think of $X$ and $Y$ as random variables drawn uniformly from $F$ we want to bound $\Pr[X * Y = f(X) \oplus g(Y)]$. This is a difficult task because it is difficult to approximate the mixed term $X * Y$ as a sum of terms that depend on $X$ or $Y$ alone. At the moment, we can show a bound of $2^{-n/2}$ by defining a family of events $E_{y} \iff X * y = f(X) \oplus g(y)$ and noticing that these events for $y \neq z$ are fairly incompatible, more specifically $\Pr[E_{y} \wedge E_{z}] \leq 2^{-n}$.
Now, however, we are interested in a more difficult game in which there are $m$ random variables: $X_{1}, X_{2}, \ldots, X_{m}$ and we want to approximate the product $X_{1} * X_{2} * \ldots * X_{m}$ as a sum of $m$ terms such that the $k$-th term depends on all the random variables except for the $k$-th one. In other words, we need to bound
\begin{equation} \Pr[X_{1} * X_{2} * \ldots * X_{m} = \bigoplus_{k = 1}^{m} f_{k}(X_{1}, X_{2}, \ldots, X_{k - 1}, X_{k + 1}, \ldots, X_{m} )]. \end{equation}
We believe that this should still be difficult but we do not know how to show this.
Alternatively, you can think of a polynomial $P(X_{1}, X_{2}, \ldots, X_{m})$, which has the property that the only term which contains all $m$ variables takes the form $X_{1} * X_{2} * \ldots * X_{m}$. Otherwise, the polynomial is arbitrary. How many zeroes can it have?
Any suggestions or references would be very appreciated.