  The main result of this paper is a coefficient formula that sharpens and
generalizes Alon and Tarsi's Combinatorial Nullstellensatz, which provides some
information about the polynomial map $P|_{\X_1\times...\times\X_n}$ when only
incomplete information about the polynomial $P(X_1,...c,X_n)$ is given.
  In a very general working frame, the grid points
$x\in\X_1\times\...b\times\X_n$ which do not vanish under an algebraic solution
-- a certain describing polynomial $P(X_1,...c,X_n)$ -- correspond to the
explicit solutions of a problem. As a consequence of the coefficient formula,
we prove that the existence of an algebraic solution is equivalent to the
existence of a nontrivial solution to a problem. By a problem, we mean
everything that "owns" both, a set $\S$, which may be called the \emph{set of
solutions}; and a subset $\St\subseteq\S$, the \emph{set of trivial solutions}.
  We give several examples of how to find algebraic solutions, and how to apply
our coefficient formula. These examples are mainly from graph theory and
combinatorial number theory, but we also prove several versions of Chevalley
and Warning's Theorem, including a generalization of Olson's Theorem, as
examples and useful corollaries.
  We obtain a permanent formula by applying our coefficient formula to the
matrix polynomial, which is a generalization of the graph polynomial. This
formula is an integrative generalization and sharpening of:
  1. Ryser's permanent formula. 2. Alon's Permanent Lemma. 3. Alon and Tarsi's
Theorem about orientations and colorings of graphs.
  Furthermore, in combination with the Vigneron-Ellingham-Goddyn property of
planar (n)regular graphs, the formula contains as very special cases:
  4. Scheim's formula for the number of edge (n)-colorings of such graphs. 5.
Ellingham and Goddyn's partial answer to the list coloring conjecture.
