One considers a formal power series ring \(k[[X_ 1,...,X_ n]]\) over a field k. One puts an admissible order on the set of all terms T (which is a well-ordering of T!, following from Dickson’s lemma). Let LT(F) denote the least element of a finite set F. The Hironaka theorem summarizes the most important facts, known about standard bases. As one passes from formal power series to polynomials, then the Hironaka theorem becomes, mutatis mutandis, a summary of the basic facts of Gröbner bases.
This paper adds some results, known for Gröbner bases, to the theory of standard bases. - First of all, the stability theorem for Gröbner bases (i.e. the property of a finite set of polynomials to be a Gröbner basis is locally stable in the topological space of admissible term orders), is “translated” into an analogous statement for standard bases. - Then the critical-pair-criterion, known for Gröbner bases, is carried over into a criterion for a finite set of power series to be a standard basis of the ideal it generates.
As all these results are non-algorithmic, it is natural to question the computability. The author concludes by discussing this point.