  We propose a new efficient algorithm to obtain the locally minimal generating
set of the syzygies for an ideal, i.e. a generating set whose proper subsets
cannot be generating sets. Syzygy is a concept widely used in the current study
of scattering amplitudes. This new algorithm can deal with more syzygies
effectively because a new generation of syzygies is obtained in each step and
the irreducibility of this generation is also verified in the process. This
efficient algorithm can also be applied in getting the syzygies for the
modules. We also show a typical example to illustrate the potential application
of this method in scattering amplitudes, especially the Integral-By-Part(IBP)
relations of the characteristic two-loop diagrams in the Yang-Mills theory.
