ABSTRACT:
The process of computing a Gr?bner basis may involve large
numbers of intermediate coefficients, even when the final Gr?bner basis does not
involve large coefficients.Shirayanagi proposed a new approach based on floating point arithmetic in the case K is
a subfield of the real numbers [20]. Basically he mimiced Buchberger’s algorithm. However, the
big question then would be ”How small must floating point coefficients be to be considered zero?”.
For this purpose, Shirayanagi can construct the floating point approximation
Gr?bner basis. First, we should guess the support of the desired Gr?bner basis from a floating
point Gr?bner basis. Next, assuming this support to be correct, then, we use linear algebra,
namely, the method of indeterminate coefficients to determine the exact values for the coefficients.
Related work includes the FGLM algorithm and its modular version. This method is new in the
sense that it can be considered as a floating point approach to the linear algebra method. The
result of Maple computing experiments indicate that this method can be very effective in the case
of nonrational coefficients, especially the ones including transcendental constants.
Finally, it is well known that in the computation of Gr?bner bases an arbitrary 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.
Faugère and Liang showed how such phenomenon may be detected and even repaired by adding a
new variable and a binomial relation each time. Their main algorithm, named TSVn corresponding to Buchberger’s algorithm,
can compute more stable Gr?bner bases of equivalent ideals (with the same set of zeros) and thus
are suitable for the computation of Gr?bner bases for ideals generated by polynomials with floating
point coefficients. The main theorem of this part is that any monomial basis (containing 1) of
the quotient ring can be found using TSV strategy.