## Algorithmic Refinement of an Approximate Basis of a Zero-dimensional Polynomial Ideal

Hans J. Stetter (Tech. University, Vienna)

Approximate (floating-point) basis computation for the ideal
generated by a multivariate polynomial system Q, where Q is
a complete intersection, suffers from the
following drawback: For s variables where s>1,
ideal bases (e.g. Groebner bases)
generally have more than s elements; a generic perturbation of the
coefficients as in an approximate computation makes the basis
inconsistent. While this does not critically affect some tasks (e.g.
computation of approximate zeros), it is a principal obstacle in others.

We explain a floating-point algorithm which refines the coefficients of an
inconsistent approximate border basis
with an associated
normal set (e.g. a Groebner basis) into the exact
border basis of , except for
linearization and round-off effects. (If necessary, these can be made
arbitrarily small by iteration and sufficient precision resp.) This
algorithm opens the way for a more wide-spread use of approximate
computation in polynomial algebra.

A crucial ingredient of the algorithm which is intesting and important on its own, is the determination of a minimal basis for the syzygies of a border basis.