Standard bases and some computations in rings of power series.

*(English)*Zbl 0709.13013Let \(K[[X_ 1,...,X_ n]]\) be the ring of formal power series over a field K in n indeterminates. Let T be the set of all terms in \(X_ 1,...,X_ n\) and denote by \(<\) an admissible term order. For every power series f, and every term t, c(t,f) denotes the coefficient of t in f; T(f) stands for all terms in f and LT(f) is the least element of T(f) with respect to \(<\). For a subset of \(K[[X_ 1,...,X_ n]]\), \(LT(S)=LT(f)| f\in S\). The author points out that there is no general proof for Hironaka’s theorem for both arbitrary field and arbitrary admissible order. The Hironaka theorem goes as follows:

Let \(I\subset K[[X_ 1,...,X_ n]]\), an ideal. Then there exists a finite set \(S\subset I\) such that for every f in I, there is g in S with LT(g)\(| LT(f)\). Any such S is a basis of I (called a standard basis). For any power series f and any standard basis \(S=g_ 1,...,g_ m\) of I, there exists a unique power series r such that \((1)\quad f=\sum^{m}_{i=1}g_ iq_ i+r\) \((q_ i\in K[[X_ 1,...,X_ n]]\); \((2)\quad for\) all \(s\in LT(S)\), \(t\in T(r):s| t.\)

The author also states that some definitions of a standard basis are equivalent in case of formal power series rings. The proof of both results are given in the paper. The method of reduction of polynomials by an ideal basis is adapted to this case, leading to a proof by \(\lambda\)- induction. The main interest is in admissible orders, not being of type \(\omega\). The author entirely focuses on the lexicographical order. It is explained what “effective calculation” stands for in this case.

The main part of the paper deals with a special case: division by a principal ideal, so that the problem of calculation a standard basis from any basis disappears. Further, an algorithm to compute the Hironaka remainder of a power series modulo a principal ideal with respect to the lexicographical term order is given.

Let \(I\subset K[[X_ 1,...,X_ n]]\), an ideal. Then there exists a finite set \(S\subset I\) such that for every f in I, there is g in S with LT(g)\(| LT(f)\). Any such S is a basis of I (called a standard basis). For any power series f and any standard basis \(S=g_ 1,...,g_ m\) of I, there exists a unique power series r such that \((1)\quad f=\sum^{m}_{i=1}g_ iq_ i+r\) \((q_ i\in K[[X_ 1,...,X_ n]]\); \((2)\quad for\) all \(s\in LT(S)\), \(t\in T(r):s| t.\)

The author also states that some definitions of a standard basis are equivalent in case of formal power series rings. The proof of both results are given in the paper. The method of reduction of polynomials by an ideal basis is adapted to this case, leading to a proof by \(\lambda\)- induction. The main interest is in admissible orders, not being of type \(\omega\). The author entirely focuses on the lexicographical order. It is explained what “effective calculation” stands for in this case.

The main part of the paper deals with a special case: division by a principal ideal, so that the problem of calculation a standard basis from any basis disappears. Further, an algorithm to compute the Hironaka remainder of a power series modulo a principal ideal with respect to the lexicographical term order is given.

Reviewer: G.Molenbergh

##### MSC:

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |

13F25 | Formal power series rings |

68W30 | Symbolic computation and algebraic computation |

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##### References:

[1] | Briançon, J., Weierstrass préparé à la Hironaka, Astérisque, 7-8, 67-76, (1973) · Zbl 0297.32004 |

[2] | Buchberger, B., An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal (in German), () · Zbl 1158.01307 |

[3] | Buchberger, B., Gröbner bases: a criterion for the solvability of algebraic systems of equations (in German), Aequ. math., 4/3, 374-383, (1970) |

[4] | Galligo, A., A propos du théorème de préparation de Weierstrass, (), 543-579 |

[5] | Galligo, A., Théorème de division et stabilité en géométrie analytique locale, Ann. inst. Fourier, 29, 2, 107-184, (1979) · Zbl 0412.32011 |

[6] | Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. math., 79, 109-326, (1964) · Zbl 0122.38603 |

[7] | Mora, T., (), Preprint |

[8] | Robbiano, L., On the theory of graded structures, J. symb. comp., 2, 139-170, (1986) · Zbl 0609.13007 |

[9] | Späth, L., The Weierstrass preparation theorem (in German), Crelle J., 161, 95-100, (1929) |

[10] | Weispfenning, V., Admissible orders and linear forms, ACM SIGSAM bulletin, 21, 16-18, (1987) · Zbl 0655.13017 |

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