  As starting point, we formulate a corollary to the Quantitative Combinatorial
Nullstellensatz. This corollary does not require the consideration of any
coefficients of polynomials, only evaluations of polynomial functions. In
certain situations, our corollary is more directly applicable and more
ready-to-go than the Combinatorial Nullstellensatz itself. It is also of
interest from a numerical point of view. We use it to explain a well-known
connection between the sign of 1-factorizations (edge colorings) and the List
Edge Coloring Conjecture. For efficient calculations and a better understanding
of the sign, we then introduce and characterize the sign of single 1-factors.
We show that the product over all signs of all the 1-factors in a
1-factorization is the sign of that 1-factorization. Using this result in an
algorithm, we attempt to prove the List Edge Coloring Conjecture for all graphs
with up to 10 vertices. This leaves us with some exceptional cases that need to
be attacked with other methods.
