Let
be an ideal, a field,
and a monomial ordering (not necessarily a well-ordering). Let be the leading ideal of with respect to . A standard basis (in case of a well-ordering a Gröbner basis)
is defined by the property that is generated by the leading monomials
of . Usually one considers a standard basis associated to the uniquely determined minimal system of monomials generating a minimal standard basis. In case of border bases, Janet bases or Pommaret bases (usually only defined for well-orderings but we generalize the concept to any ordering) the underlying standard bases have as leading monomials special (non minimal) generators of We describe an algorithm which computes these bases using a minimal standard bases and the corresponding special generators of . We have implemented these algorithms in SINGULAR (cf. [DGPS16]) including modular and parallel implementations and give timings to compare them (cf.[KP17]). We also discuss the verification of the modular algorithm to compute border bases and Janet bases.

Key Words: Gröbner basis, standard basis, Janet Basis, Border Basis, modular computation

2010 Mathematics Subject Classification: Primary 13P99, 13E05

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