  We look at the number of solutions of an equation of the form
f_1*f_2*...*f_k=a in a finite field, where each f_i is a multilinear
polynomial. We use two methods to construct a solution of this problem for the
cases a=0, a<>0, and we generally get a semi-explicit formula. We show that
this formula can generate a more efficient algorithm than the traditional
algorithm which consists to make a systematic computation. We also give
explicit formulas in some special cases, and an application of our main result
to the search of the weight hierarchy of the multilinear code with separated
variables.
