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In the context of modeling biological systems, it is of interest to generate ideals of points with a unique reduced Gröbner basis, and the first main goal of this paper is to identify classes of ideals in polynomial rings which share this property. Moreover, we provide methodologies for constructing such ideals. We then relax the condition of uniqueness. The second and most relevant topic discussed here is to consider and identify pairs of ideals with the same number of reduced Gröbner bases, that is, with the same cardinality of their associated Gröbner fan.
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This content will become publicly available on May 1, 2026
A new algorithm for Gröbner bases conversion
A new approach for Gröbner bases conversion of polynomial ideals (over a field) of arbitrary dimension is presented. In contrast to the only other approach based on Gröbner fan and Gröbner walk for positive dimensional ideals, the proposed approach is simpler to understand, prove, and implement. It is based on defining for a given polynomial, a truncated sub-polynomial consisting of all monomials that can possibly become the leading monomial with respect to the target ordering: the monomials between the leading monomial of the target ordering and the leading monomial of the initial ordering. The main ingredient of the new algorithm is the computation of a Gröbner basiswith respect tothe target ordering for the ideal generated by such truncated parts of the input Gröbner basis. This is done using the extended Buchberger algorithm that also outputs the relationship between the input and output bases. That information is used in attempts to recover a Gröbner basis of the idealwith respect tothe target ordering. In general, more than one iteration may be needed to get a Gröbner basiswith respect tothe target ordering since truncated polynomials may miss some leading monomials. The new algorithm has been implemented inMapleand its operation is illustrated using an example. The performance of this implementation is compared with the implementations of other approaches inMaple.In practice, a Gröbner basiswith respect toa target ordering can be computed in a single iteration on most examples.
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- Award ID(s):
- 1908804
- PAR ID:
- 10642525
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Journal of symbolic computation
- ISSN:
- 0747-7171
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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