The number of solutions of an equation related to a product of multilinear polynomials

We look at the number of solutions of an equation of the form f1f2...fk = a in a finite field, where each fi is a multilinear polynomial. We use two methods to construct a solution of this problem for the cases a = 0, a 6 0, and we generally get a semi-explicit formula. We show that this formula can generate a more efficient algorithm than the t 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.