Floating point Gröbner bases

Abstract

Bracket coefficients for polynomials are introduced. These are like specific precision floating point numbers together with error terms. Working in terms of bracket coefficients, an algorithm that computes a Gröbner basis with floating point coefficients is presented, and a new criterion for determining whether a bracket coefficient is zero is proposed. Given a finite set F of polynomials with real coefficients, let Gμ be the result of the algorithm for F and a precision value μ, and G be a true Gröbner basis of F. Then, as μ approaches infinity, Gμ converges to G coefficientwise. Moreover, there is a precision M such that if μ ≥ M, then the sets of monomials with non-zero coefficients of Gμ and G are exactly the same. The practical usefulness of the algorithm is suggested by experimental results.