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Floating point Gröbner bases

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  • Shirayanagi, Kiyoshi

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.

Suggested Citation

  • Shirayanagi, Kiyoshi, 1996. "Floating point Gröbner bases," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 42(4), pages 509-528.
    • Handle: RePEc:eee:matcom:v:42:y:1996:i:4:p:509-528
    DOI: 10.1016/S0378-4754(96)00027-4
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