46 - Zacharias
from PART SEVEN - Beyond
Published online by Cambridge University Press: 05 April 2016
Summary
In her 1978 Bachelor's thesis, Zacharias discussed how to extend Buchberger's theory and algorithm from the case of a polynomial ring over a field (as presented in the second volume) to that of polynomials over a Noetherian ring. In the introduction she wrote:
Since the structure of R[X1, …, Xn] is totally determined by R, any problem in R[X1, …, Xn] is in effect solved by translating it to some equivalent problem in R. In the recursive approach we take the problem and translate it into R[X1, …, Xn−1], then R[X1, …, Xn−2], and so on, until we get to R and finally start solving it. It is apparent that there are many opportunities for unnecessary work in this project. There is also an advantage in going straight from R[X1, …, Xn] to R from a theoretical point view. For such a direct approach might make it easier to discern the relationship between the coefficient ring R and the structure it imposes on R[X1, …, Xn].
Her approach is based on the remark that, if R[X1, …, Xn] satisfies an idealtheoretical property, the same property must also be satisfied by R and thus effectiveness of a such property necessarily must be assumed in R and thus can be used as a seed for a procedure that effects such a property in R[X1, …, Xn]. In particular, the aim of Buchberger's theory being membership testing and syzygies computations, such properties can be assumed in R as a tool for defining and computing Gröbner bases.
In her approach she continued the approach of Szekeres, who in 1952 studied the structure of ideal bases for univariate polynomials and the extension performed in 1974 by Richman, which
generalized the construction to coefficient rings in which ideal membership and syzygies are solvable. He then used this construction to show that ideal membership and syzygies are solvable in the polynomial ring as well. Then by induction he extended his results to multivariate polynomial rings.
Her main contribution is removing the useless inductive approach, thus making her tools available in the more general setting of monoid rings.
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- Solving Polynomial Equation Systems IV , pp. 3 - 99Publisher: Cambridge University PressPrint publication year: 2016