Numerical Computation of Gröbner Bases for Zero-dimensional Polynomial Ideals

It is well known that in the computation of Gröbner bases an arbitrarily small perturbation in the coefficients of polynomials may lead to a completely different staircase even if the roots of the polynomials change continuously. This phenomenon is called pseudo singularity in this paper. We show how such phenomenon may be detected and even “repaired” by adding a new variable and a binomial relation each time. To investigate how often likely pseudo singularities may happen in numerical computation of Gröbner bases, two algorithms e-Buchberger and e-MatrixF5 are provided. Our main algorithm, named VSGBn corresponding to Buchberger’s algorithm, can compute “more stable” Gröbner bases of equivalent ideals (with the same set of zeros) and thus are suitable for the computation of Gröbner bases for ideals generated by polynomials with floating-point coefficients. The main theorem of this paper is that any monomial basis (containing 1) of the quotient ring can be found out using VSGB strategy. Experiments show that the algorithms can be used to solve some non-trivial problems. Mathematics Subject Classification (2000). Primary I.1.2; Secondary F.2.1.