Uwe Schauz, Describing Polynomials as Equivalent to Explicit Solutions

We present a coefficient formula which provides some information about the polynomial map when only incomplete information about a polynomial is given. It is an integrative generalization and sharpening of several known results and has many applications, among these are:

1. The fact that polynomials in just one variable have at most deg(P) roots.
2. Alon and Tarsi’s Combinatorial Nullstellensatz.
3. Chevalley and Warning’s Theorem about the number of simultaneous zeros of systems of polynomials over finite fields.
4. Ryser’s Permanent Formula.
5. Alon’s Permanent Lemma.
6. Alon and Tarsi’s Theorem about orientations and colorings of graphs.
7. Scheim’s formula for the number of edge n-colorings of planar n-regular graphs.
8. Alon, Friedland and Kalai’s Theorem about regular subgraphs.
9. Alon and Füredi’s Theorem about cube covers.
10. Cauchy and Davenport’s Theorem from additive number theory.
11. Erdős, Ginzburg and Ziv’s Theorem from additive number theory.