An efficient algorithm for computing a comprehensive Gröbner system of a parametric polynomial system

Abstract

An algorithm to generate a minimal comprehensive Gröbner basis of a parametric polynomial system from an arbitrary faithful comprehensive Gröbner system is presented. A basis of a parametric polynomial ideal is a comprehensive Gröbner basis if and only if for every specialization of parameters in a given field, the specialization of the basis is a Gröbner basis of the associated specialized polynomial ideal. The key idea used in ensuring minimality is that of a polynomial being essential with respect to a comprehensive Gröbner basis. The essentiality check is performed by determining whether a polynomial can be covered for various specializations by other polynomials in the associated branches in a comprehensive Gröbner system. The algorithm has been implemented and successfully tried on many examples from the literature. Keywordscomprehensive Gröbner basis, minimal comprehensive Gröbner basis, parametric polynomial system, specialization of polynomial systems, essentiality.

An Algorithm to Check Whether a Basis of a Parametric Polynomial System is a Comprehensive Gröbner Basis and the Associated Completion Algorithm

“…The KSW algorithm thus computes a minimal CGS. However, the CGB recovered by taking the union of these corresponding Gröbner bases fails to be minimal [8]. That is despite the fact that all segments in the KSW are disjoint whereas that need not be the case for other algorithms.…”

“…Algorithms proposed by Kapur, Sun and Wang in [6,7] compute faithful CGSs, whereas algorithms of Suzuki and Sato [17] as well as the MCCGS [10] and Gröbner cover [12] algorithms of Montes and his collaborators do not compute faithful CGSs.…”

“…The modified algorithm computes a unique Gröbner cover for a given parametric ideal with a given admissible term ordering. It first homogenizes the given basis, then computes the canonical Gröbner Cover of this homogeneous ideal using [11,6] as the first phase, and finally maps it back to the affine space. However, Gröbner cover is not faithful, thus no CGB emerges from it.…”

“…Because of their applications, these topics have been well investigated by researchers and a number of algorithms have been proposed to construct such objects for parametric polynomial systems ( [11], [19], [15], [16], [17], [9], [4], [14], [20], [10], [12], [6], [7]). An algorithm for simultaneously generating a comprehensive Gröbner system (CGS) and a comprehensive Gröbner basis (CGB) by Kapur, Sun and Wang (KSW) [7] is particularly noteworthy because of its many nice properties: (i) fewer segments (branches) in the resulting CGS, (ii) all polynomials in the CGS and CGB are faithful meaning that they are in the input ideal, and more importantly, (iii) the algorithm has been found efficient in practice [13].…”

“…However, this algorithm turns out to be not easy to design. another indirect algorithm [8]. Section 7 makes concluding remarks and discusses topics for future research.…”