We present a coefficient formula which provides some formation about the polynomial map $P|_{I_1\times\dotsb\times I_n}|$ when only incomplete information about the polynomial $P(X_1, \dotsc, X_n)$ is given. It is an integrative generalization and sharpening of several known results and has many applications, among these are:\vspace{-1ex} \begin{enumerate}\setlength{\itemsep}{-0.5ex} \item The fact that polynomials $P(X_1)\neq0$ in just one variable have at most $\deg(P)$ roots. \item Alon and Tarsi's Combinatorial Nullstellensatz. \item Chevalley and Warning's Theorem about the number of simultaneous zeros of systems of polynomials over finite fields. \item Ryser's Permanent Formula. \item Alon's Permanent Lemma. \item Alon and Tarsi's Theorem about orientations and colorings of graphs. \item Scheim's formula for the number of edge \(n\)-colorings of planar \(n\)-regular graphs. \item Alon, Friedland and Kalai's Theorem about regular subgraphs. \item Alon and F\"uredi's Theorem about cube covers.