Bideterminants, arborescences and extension of the Matrix-Tree Theorem to semirings

The Matrix-Tree Theorem is a well-known combinatorial result relating the value of the miners of a certain square matrix to the sum of the weights of the arborescences (= rooted directed trees) in the associated graph. We prove an extension of this result to algebraic structures much more general than the field of real numbers, namely commutative semirings. In such structures, the first law (addition) is not assumed to be invertible, therefore the combinatorial proof given here significantly differs from earlier proofs for the standard case. In particular, it requires the use of the concept of bideterminant of a matrix, an extension of the classical concept of determinant.