Matchings in Groups and Vector Spaces

A model for class of bipartite graphs is introduced, and a certain type of perfect matching, called an acyclic matching, is defined to imply a nonvanishing determinant for a class of weighted biadjacency matrices. Using the existence results in matching theory, an old linear algebra problem on the removable sets of monomials from a generic homogenous polynomial through linear changes in its variables is discussed. The matching property defined in Zn is first generalized to abelian groups and later to arbitrary groups. The local matching property is defined to give alternative proofs for existing results in matching theory. Linear analogues of results concerning matchings in groups are formulated and proven. Finally, an improvement of the fundamental theorem of algebra is given as an application of matchings in field extensions.