We generalize two theorems of Alon, known as Combinatorial Nullstellensatz, about polynomials in n variables over integral domains of characteristic zero, vanishing on certain sets of points, by including the possibility of ?multiple vanishings?, i.e., vanishings of certain compositions of partial derivatives of the polynomial in question. The proof that we give for the first theorem is a direct one, by using a triple induction. We then use the first theorem in the proof of the second one.

combinatorial Nullstellensatz, polynomials in n variables, partial derivatives.