We first present purely combinatorial proofs of two facts: the well-known
fact that a monomial ordering must be a well ordering, and the fact (obtained
earlier by Buchberger, but not widely known) that the division procedure in
the ring of multivariate polynomials over a field terminates even if the
division term is not the leading term, but is freely chosen. The latter is
then used to introduce a previously unnoted, seemingly weaker, criterion for
an ideal basis to be Grobner, and to suggest a new heuristic approach to
Grobner basis computations.
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.
Abstract
The displacement analysis problem for planar mechanisms can be written as a system of algebraic equations, in particular as a system of multivariate polynomial equations. Elimination theory based on resultants and polynomial continuation are some of the methods which have been used to solve this problem. This paper explores an alternate approach, based on Gröbner bases, to solve the displacement analysis problem for planar mechanisms. It is shown that the reduced set of generators obtained using the Buchberger’s algorithm for Gröbner bases not only yields the input-output polynomial for the mechanism, but also provides comprehensive information on the number of closures and the relationships between various links of the mechanism. Numerical examples illustrating the applicability of Gröbner bases to displacement analysis of 10 and 12-link mechanisms and determination of coupler curve equation for 8-link mechanisms are presented.
This paper deals with a generalization of the secret sharing using Chinese remainder theorem over the integers to multivariate polynomials over a finite field. We work with the ideals and their Gröbner bases instead of integer moduli. Therefore, the proposed method is called GB secret sharing. It was initially presented in our previous paper. Now we prove that any threshold structure has ideal GB realization. In a generic threshold modular scheme in ring of integers the sizes of the share space and the secret space are not equal. So, the scheme is not ideal and our generalization of modular secret sharing to the multivariate polynomial ring is more secure.
We introduce methods that use Grobner bases for secure secret sharing schemes. The description is based on polynomials in the ring R = K[X1, . . . , Xl] where identities of the participants and shares of the secret are or are related to ideals in R. Main theoretical results are related to algorithmical reconstruction of a multivariate polynomial from such shares with respect to given access structure, as a generalisation of classical threshold schemes. We apply constructive Chinese remainder theorem in R of Becker and Weispfenning. Introduced ideas find their detailed exposition in our related works
A note on multivariate polynomial division and Gröbner bases