Multivariate Resultants in Bernstein Basis (Book Chapter)
In T. Sturm and C. Zengler (Eds.), "Automated Deduction in Geometry," Springer Verlag, 60- 85, August 2011
Manfred Minimair, Ph.D.
Department of Mathematics and Computer Science
Macaulay and Dixon resultant formulations are proposed for parametrized multivariate polynomial systems represented in Bernstein basis. It is proved that the Macaulay resultant for a polynomial system in Bernstein basis vanishes for the total degree case if and only if the either the polynomial system has a common Bernstein-toric root, a common infinite root, or the leading forms of the polynomial system obtained by replacing every variable xi in the original polynomial system by yi/1+yi have a non-trivial common root. For the Dixon resultant formulation, the rank sub-matrix constructions for the original system and the transformed system are shown to be essentially equivalent. Known results about exactness of Dixon resultants of a sub-class of polynomial systems as discussed in Chtcherba and Kapur in Journal of Symbolic Computation (August, 2003) carry over to polynomial systems represented in the Bernstein basis. Furthermore, in certain cases, when the extraneous factor in a projection operator constructed from the Dixon resultant formulation is precisely known, such results also carry over to projection operators of polynomial systems in the Bernstein basis where extraneous factors are precisely known. Applications of these results in the context of geometry theorem proving, implicitization and intersection of surfaces with curves are discussed. While Macaulay matrices become large when polynomials in Bernstein bases are used for problems in these applications, Dixon matrices are roughly of the same size.