Dingkang Wang

A new algorithm for computing comprehensive Gröbner systems

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 Satos algorithm as well as Nabeshimas algorithm, resulting in considerable efficiency. As a result, the algorithm is able to compute comprehensive Gröbner 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 Weispfennings 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 Kalkbreners 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 Gröbner 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 Rabinovitchs trick using a new variable as in Nabeshimas 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-Satos algorithm and Nabeshimas speed-up algorithm. The algorithm has been successfully used to solve the famous P3P problem from computer vision.

Curve fitting and optimal interpolation on CNC machines based on quadratic B-splines

In this paper, curve fitting of 3-D points generated by G01 codes and interpolation based on quadratic B-splines are studied. Feature points of G01 codes are selected using an adaptive method. Next, quadratic B-splines are obtained as the fitting curve by interpolating these feature points. Computations required in implementing the velocity planning algorithm mentioned by Timer et al. are very complicated because of the appearance of high-order curves. An improved time-optimal method for the quadratic B-spline curves is presented to circumvent this issue. The algorithms are verified with simulations as well as on real CNC machines.

A generalized criterion for signature related Gröbner basis algorithms

A generalized criterion for signature related algorithms to compute Gröbner basis is proposed in this paper. Signature related algorithms are a popular kind of algorithms for computing Gröbner basis, including the famous F5 algorithm, the F5C algorithm, the extended F5 algorithm and the GVW algorithm. The main purpose of current paper is to study in theory what kind of criteria is correct in signature related algorithms and provide a generalized method to develop new criteria. For this purpose, a generalized criterion is proposed. The generalized criterion only relies on a general partial order defined on a set of polynomials. When specializing the partial order to appropriate specific orders, the generalized criterion can specialize to almost all existing criteria of signature related algorithms. For admissible partial orders, a proof is presented for the correctness of the algorithm that is based on this generalized criterion. And the partial orders implied by the criteria of F5 and GVW are also shown to be admissible in this paper. More importantly, the generalized criterion provides an effective method to check whether a new criterion is correct as well as to develop new criteria for signature related algorithms.

The F5 algorithm in Buchberger’s style

The famous F5 algorithm for computing Gröbner basis was presented by Faugère in 2002. The original version of F5 is given in programming codes, so it is a bit difficult to understand. In this paper, the F5 algorithm is simplified as F5B in a Buchberger’s style such that it is easy to understand and implement. In order to describe F5B, we introduce F5-reduction, which keeps the signature of labeled polynomials unchanged after reduction. The equivalence between F5 and F5B is also shown. At last, some versions of the F5 algorithm are illustrated.

A signature-based algorithm for computing Gröbner bases in solvable polynomial algebras

Signature-based algorithms, including F5, F5C, G2V and GVW, are efficient algorithms for computing Gröbner bases in commutative polynomial rings. In this paper, we present a signature-based algorithm to compute Gröbner bases in solvable polynomial algebras which include usual commutative polynomial rings and some non-commutative polynomial rings like Weyl algebra. The generalized Rewritten Criterion (discussed in Sun and Wang, ISSAC 2011) is used to reject redundant computations. When this new algorithm uses the partial order implied by GVW, its termination is proved without special assumptions on computing orders of critical pairs. Data structures similar to F5 can be used to speed up this new algorithm, and Gröbner bases of syzygy modules of input polynomials can be obtained from the outputs easily. Experimental data show that most redundant computations can be avoided in this new algorithm.

On the automatic derivation of a set of geometric formulae

Leta, b, andc be the three sides of a triangleABC, ai,bi,ci andae,be, ce be the lengths of the three internal and external bisectors of the three anglesA, B, andC respectively. It is easy to express the bisectors as formulae of the sides. In this paper, we solve a problem proposed by H. Zassenhaus: for any three different bisectors in {ai, bi, ci, ae, be, ce}, finding the relations between each side of the triangle and the three chosen bisectors. We also prove that given any general values for three different bisectors (internal or external) of a triangle, we can not draw the triangle using a ruler and a pair of compasses alone. The formulae mentioned above are derived automatically using a general method of mechanical formula derivation.

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

A new efficient algorithm for computing a comprehensive Grobnersystem of a parametric polynomial ideal over k[U][X] is presented. This algorithm generates fewer branches (segments) compared to previously proposed algorithms including Suzuki and Satos algorithm as well as Nabeshimas algorithm. As a result, the algorithm is able to compute comprehensive Grobnersystems 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 Weispfennings algorithm with a key insight by Suzuki and Sato who proposed computing first a Grobnerbasis of an ideal over k[U,X] before performing any branches based on parametric constraints. The proposed algorithm exploits the result that along any branch in a tree corresponding to a comprehensive Grobnersystem, 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 Rabinovitchs trick using a new variable as in Nabeshimas algorithm. The proposed algorithm has been implemented in Magma and Singular, and experimented with a number of examples from different applications. Its performance (the number of branches and execution time) has been compared with several other existing algorithms. A number of heuristics and efficient checks have been incorporated into the Magma implementation, especially in the case when the ideal of parametric constraints is 0-dimensional. The algorithm has been successfully used to solve a special case of the famous P3P problem from computer vision.

An Algorithm for Transforming Regular Chain into Normal Chain

We present an improved algorithm to compute the normal chain from a given regular chain such that their saturation ideals are the same. Our algorithm is based on solving a system of linear equations and the original method computes the resultants of multivariate polynomials. From the experiments, for the random polynomials, our algorithm is much more efficient than the original one.

Computing comprehensive Gröbner systems and comprehensive Gröbner bases simultaneously

In Kapur et al (ISSAC, 2010), a new method for computing a comprehensive Grobner system of a parameterized polynomial system was proposed and its efficiency over other known methods was effectively demonstrated. Based on those insights, a new approach is proposed for computing a comprehensive Grobner basis of a parameterized polynomial system. The key new idea is not to simplify a polynomial under various specialization of its parameters, but rather keep track in the polynomial, of the power products whose coefficients vanish; this is achieved by partitioning the polynomial into two parts-nonzero part and zero part for the specialization under consideration. During the computation of a comprehensive Grobner system, for a particular branch corresponding to a specialization of parameter values, nonzero parts of the polynomials dictate the computation, i.e., computing S-polynomials as well as for simplifying a polynomial with respect to other polynomials; but the manipulations on the whole polynomials (including their zero parts) are also performed. Grobner basis computations on such pairs of polynomials can also be viewed as Grobner basis computations on a module. Once a comprehensive Grobner system is generated, both nonzero and zero parts of the polynomials are collected from every branch and the result is a faithful comprehensive Grobner basis, to mean that every polynomial in a comprehensive Grobner basis belongs to the ideal of the original parameterized polynomial system. This technique should be applicable to other algorithms for computing a comprehensive Grobner system as well, thus producing both a comprehensive Grobner system as well as a faithful comprehensive Grobner basis of a parameterized polynomial system simultaneously. The approach is exhibited by adapting the recently proposed method for computing a comprehensive Grobner system in (ISSAC, 2010) for computing a comprehensive Grobner basis. The timings on a collection of examples demonstrate that this new algorithm for computing comprehensive Grobner bases has better performance than other existing algorithms.

On computing Gröbner bases in rings of differential operators

Insa and Pauer presented a basic theory of Gröbner basis for differential operators with coefficients in a commutative ring in 1998, and a criterion was proposed to determine if a set of differential operators is a Gröbner basis. In this paper, we will give a new criterion such that Insa and Pauer’s criterion could be concluded as a special case and one could compute the Gröbner basis more efficiently by this new criterion.