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**Gröbner bases: an algorithmic method in polynomial ideal theory.**
*(English)*
Zbl 0587.13009

Multidimensional systems theory, Progress, directions and open problems, Math. Appl., D. Reidel Publ. Co. 16, 184-232 (1985).

[For the entire collection see Zbl 0562.00017.]

From the author’s introduction: ”Problems connected with ideals generated by finite sets \(F\) of multivariate polynomials occur, as mathematical subproblems, in various branches of system theory,... The method of Gröbner bases is a technique that provides algorithmic solutions to a variety of such problems, for instance, exact solutions of \(F\) viewed as a system of algebraic equations, computations in the residue class ring modulo the ideal generated by \(F\), decision about various properties of the ideal generated by \(F\), polynomial solution of the linear homogeneous equations with coefficients in \(F\), word problems modulo ideals and in commutative semigroups (reversible Petri nets), bijective enumeration of all polynomial ideals over a given coefficient domain etc.”

The article under review gives a survey of the method of Gröbner bases and presents an improved version of their algorithmic construction. The author discusses applications as indicated above. One section deals with Gröbner bases for polynomial ideals over the integers, and the final section gives some information on implementations of the algorithm. While proofs are replaced by references to the original literature, many instructive examples illustrate the power of the algorithm.

From the author’s introduction: ”Problems connected with ideals generated by finite sets \(F\) of multivariate polynomials occur, as mathematical subproblems, in various branches of system theory,... The method of Gröbner bases is a technique that provides algorithmic solutions to a variety of such problems, for instance, exact solutions of \(F\) viewed as a system of algebraic equations, computations in the residue class ring modulo the ideal generated by \(F\), decision about various properties of the ideal generated by \(F\), polynomial solution of the linear homogeneous equations with coefficients in \(F\), word problems modulo ideals and in commutative semigroups (reversible Petri nets), bijective enumeration of all polynomial ideals over a given coefficient domain etc.”

The article under review gives a survey of the method of Gröbner bases and presents an improved version of their algorithmic construction. The author discusses applications as indicated above. One section deals with Gröbner bases for polynomial ideals over the integers, and the final section gives some information on implementations of the algorithm. While proofs are replaced by references to the original literature, many instructive examples illustrate the power of the algorithm.

Reviewer: Winfried Bruns

### MSC:

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |

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

13A15 | Ideals and multiplicative ideal theory in commutative rings |

68W30 | Symbolic computation and algebraic computation |