Janet-Like Gröbner Bases

Abstract

We define a new type of Grobner bases called Janet-like, since their properties are similar to those for Janet bases. In particular, Janet-like bases also admit an explicit formula for the Hilbert function of polynomial ideals. Cardinality of a Janet-like basis never exceeds that of a Janet basis, but in many cases it is substantially less. Especially, Janet-like bases are much more compact than their Janet counterparts when reduced Grobner bases have “sparce” leading monomials sets, e.g., for toric ideals. We present an algorithm for constructing Janet-like bases that is a slight modification of our Janet division algorithm. The former algorithm, by the reason of checking not more but often less number of nonmultiplicative prolongations, is more efficient than the latter one.