Algebraic aspects of when and how a Feynman diagram reduces to simpler ones

The method of Symmetries of Feynman Integrals defines for any Feynman diagram a set of partial differential equations. On some locus in parameter space the equations imply that the diagram can be reduced to a linear combination of simpler diagrams. This paper provides a systematic method to determine this locus and the associated reduction through an algebraic method involving factorization of maximal minors.