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**An algorithm to calculate the kernel of certain polynomial ring homomorphisms.**
*(English)*
Zbl 0860.68062

The Buchberger algorithm can calculate a finite basis for the kernel of a ring homomorphism \(\pi:K_x\to K_y\) applying it over a polynomial the ring \(K_{x,y}\). Nevertheless, the algorithm complexity grows very fast with the increase in the number of variables. In this paper, the authors propose an alternative algorithm to compute the kernel of a polynomial map assuming that the map takes monomials into monomials. The starting point for the alternative algorithm was that in some situations it seems more reasonable to compute the Gröbner basis over \(K_x\) instead of over \(K_{x,y}\), as over the former there is a larger number of variables.

The paper presents a brief review of Gröbner basis computation over \(K_{x,y}\), illustrating with examples. Following that, the solution proposed is fully discussed. The algorithm to compute the kernel of \(\pi\) is summarized and a couple of examples are given to show how it works. The complexity of the algorithm is studied and it is shown that it requires the determination of at most \(\lfloor{1\over 2}n\rfloor\) Gröbner basis over \(K_x\), whereas the Buchberger algorithm has a higher complexity as it computes the Gröbner basis over \(K_{x,y}\), which has more variables. The performance of the algorithm was investigated taking 500 random examples on a NeXT computer with 25 MHz and 20 Mb of RAM, making use of the Mathematica and PARI computer algebra systems. The results indicate that the proposed algorithm has a significant decrease in running time and additionally requires much less memory. The performance results are displayed in a table for both methods. The method proposed is useful, but as the authors comment it is limited as it can only be applied to a map \(\pi\) that sends monomials into monomials.

The paper presents a brief review of Gröbner basis computation over \(K_{x,y}\), illustrating with examples. Following that, the solution proposed is fully discussed. The algorithm to compute the kernel of \(\pi\) is summarized and a couple of examples are given to show how it works. The complexity of the algorithm is studied and it is shown that it requires the determination of at most \(\lfloor{1\over 2}n\rfloor\) Gröbner basis over \(K_x\), whereas the Buchberger algorithm has a higher complexity as it computes the Gröbner basis over \(K_{x,y}\), which has more variables. The performance of the algorithm was investigated taking 500 random examples on a NeXT computer with 25 MHz and 20 Mb of RAM, making use of the Mathematica and PARI computer algebra systems. The results indicate that the proposed algorithm has a significant decrease in running time and additionally requires much less memory. The performance results are displayed in a table for both methods. The method proposed is useful, but as the authors comment it is limited as it can only be applied to a map \(\pi\) that sends monomials into monomials.

Reviewer: W.L.Roque (Porto Alegre)

### MSC:

68W30 | Symbolic computation and algebraic computation |

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

68Q25 | Analysis of algorithms and problem complexity |

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\textit{F. Di Biase} and \textit{R. Urbanke}, Exp. Math. 4, No. 3, 227--234 (1995; Zbl 0860.68062)

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