A new algorithm for computing comprehensive Gröbner systems

Abstract

A new algorithm for computing a comprehensive Gröbner system of a parametric polynomial ideal over k [U ][X] is presented. This algorithm generates fewer branches (segments) compared to Suzuki and Sato's algorithm as well as Nabeshima's algorithm, resulting in considerable efficiency. As a result, the algorithm is able to compute comprehensive Gröb-ner systems of parametric polynomial ideals arising from applications which have been beyond the reach of other well known algorithms. The starting point of the new algorithm is Weispfenning's algorithm with a key insight by Suzuki and Sato who proposed computing first a Gröbner basis of an ideal over k[U, X] before performing any branches based on parametric constraints. Based on Kalkbrener's results about stability and specialization of Gröbner basis of ideals, the proposed algorithm exploits the result that along any branch in a tree corresponding to a comprehensive Gröb-ner system, it is only necessary to consider one polynomial for each nondivisible leading power product in k(U ) [X] with the condition that the product of their leading coefficients is not 0; other branches correspond to the cases where this product is 0. In addition, for dealing with a disequality parametric constraint, a probabilistic check is employed for radical membership test of an ideal of parametric constraints. This is in contrast to a general expensive check based on Rabinovitch's trick using a new variable as in Nabeshima's algorithm. The proposed algorithm has been implemented in Magma and experimented with a number of examples from different applications. Its performance (vis a vie number of branches and execution timings) has been compared with the Suzuki-Sato's algorithm and Nabeshima's speed-up algorithm. The algorithm has been successfully used to solve the famous P3P problem from computer vision.

Automatic deduction in (dynamic) geometry: Loci computation

“…Since then, many improvements to this theory have been made (e.g. [39], [40], [41], [18], [42], to mention a few). Although most of the proposed algorithms can be used to solve the problems raised here, in this paper we primarily use the GröbnerCover algorithm (GC) proposed in [18], deferring a thorough study of other algorithms to a future work.…”

“…Using GC authors' own words, the main focus [ [18, p. 1392]. Although the algorithm in [42] (which substantially improves the Suzuki-Sato algorithm (SS)) is generally more efficient, GC offers a compact discussion of the system, in general canonical. However, compactness in the description comes at the price of using sometimes regular functions instead of polynomials (see [18, p. 1394]).…”

Software Using the Gröbner Cover for Geometrical Loci Computation and Classification

“…Its existence was proved by Wibmer's Theorem [6], and the method and algorithms were developed in [5]. Montes implemented in Singular the grobcov.lib library [7], whose actual version incorporates Kapur-Sun-Wang algorithm [2] for computing the initial Gröbner System used in grobcov algorithm, as described in [4], and recently also the Locus algorithm described here. A more detailed description can be seen in [1].…”

Computing the Canonical Representation of Constructible Sets

“…Using Kapur-Sun-Wang algorithm [11], 329 the parameter space is divided into 11 disjoint segments S 1 ,...,S 11 , and for each segment S i a basis B i specializing Table 1 Segments and bases of Example 2 V(a 1 , a 0 , −b 0 c 1 + c 0 b 1 )\V(b 1 , a 1 , b 0 , a 0 V(−b 0 c 1 + c 0 b 1 , −a 0 c 1 + c 0 a 1 , −a 0 b 1 + b 0 a 1 )\V(a 1 , a 0 , −b 0 c 1 + c 0 b 1 V(c 1 , b 1 , a 1 , c 0 , b 0 , a 0 )\V (1) to the reduced Gröbner basis on the whole segment is given. Table 1 gives the sets S i and B i .…”