Abstract
We discuss computation of Gröbner bases using approximate arithmetic for coefficients. We show how certain considerations of tolerance, corresponding roughly to absolute and relative error from numeric computation, allow us to obtain good approximate solutions to problems that are overdetermined. We provide examples of solving overdetermined systems of polynomial equations. As a secondary feature we show handling of approximate polynomial GCD computations, using benchmarks from the literature.
Highlights
Grobner bases provide a means for solving a myriad of problems in computational algebra
The advantages of approximate arithmetic are several. First is that it avoids intermediate coefficient swell one often observes in exact Grobner basis computations
We provide several nontrivial examples from the literature on numerical polynomial system solving and approximate GCD computation to illustrate the merit of this work
Summary
Grobner bases provide a means for solving a myriad of problems in computational algebra. This family of methods attempts to stabilize the set of head terms locally, that is, in a neighborhood of a set of coefficient values that give a “nongeneric” basis skeleton They seem to work well for handling numeric conditioning problems that might arise from having polynomial leading terms with relatively small coefficients, but it is not clear whether they can be used in the case of an overdetermined system. The generalized normal form approach of [16] uses a similar method to [9, 13, 14] for handling small leading coefficients, but has access to more possibilities since the monomial ordering need not be fixed from the start These local stabilization methods seem to work best in the case of zero dimensional ideals with the same degree (number of solutions). We provide several nontrivial examples from the literature on numerical polynomial system solving and approximate GCD computation to illustrate the merit of this work
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