Exact resultants for corner-cut unmixed multivariate polynomial systems using the dixon formulation

Structural conditions on the support of a multivariate polynomial system are developed for which the Dixon-based resultant methods compute exact resultants. For cases when this cannot be done, an upper bound on the degree of the extraneous factor in the projection operator can be determined a priori, thus resulting in quick identification of the extraneous factor in the projection operator. (For the bivariate case, the degree of the extraneous factor in a projection operator can be determined a priori.) The concepts of a corner-cut support and almost corner-cut support of an unmixed polynomial system are introduced. For generic unmixed polynomial systems with corner-cut and almost corner-cut supports, the Dixon based methods can be used to compute their resultants exactly. These structural conditions on supports are based on analyzing how such supports differ from box supports of n-degree systems for which the Dixon formulation is known to compute the resultants exactly. Such an analysis also gives a sharper bound on the complexity of resultant computation using the Dixon formulation in terms of the support and the mixed volume of the Newton polytope of the support. These results are a direct generalization of the authors ’ results on bivariate systems including the results of Zhang and Goldman as well as of Chionh for generic unmixed bivariate polynomial systems with corner-cut supports

Conditions for Exact Resultants using the Dixon Formulation

Introduction A structural criteria on multivariate polynomial systems is developed such that the generalized Dixon formulation of multivariate resultants [6, 12] as well as the associated sparse resultant construction recently obtained by the authors from this formulation [5] computes the resultant exactly, i.e. these constructions do not produce any extraneous factors in the resultant computations for such polynomial systems. This result is of considerable signicance since extraneous factors arising in resultant computations of polynomial systems is a key problem faced when any multivariate resultant method for simultaneously eliminating many variables is used for elimination in a variety of applications including computer vision, robotics and kinematics, control theory, solid and geometric modeling, geometry theorem proving, biology, etc. A long-term goal of this research is to be able to identify subsets of monomials in relation to the supports of polynomials in a polynom

Cayley-Dixon resultant matrices of multi-univariate composed polynomials

Abstract. The behavior of the Cayley-Dixon resultant construction and the structure of Dixon matrices are analyzed for composed polynomial systems constructed from a multivariate system in which each variable is substituted by a univariate polynomial in a distinct variable. It is shown that a Dixon projection operator (a multiple of the resultant) of the composed system can be expressed as a power of the resultant of the outer polynomial system multiplied by powers of the leading coefficients of the univariate polynomials substituted for variables in the outer system. The derivation of the resultant formula for the composed system unifies all the known related results in the literature. A new resultant formula is derived for systems where it is known that the Cayley-Dixon construction does not contain any extraneous factors. The approach demonstrates that the resultant of a composed system can be effectively calculated by considering only the resultant of the outer system.

Cayley–Dixon projection operator for multi-univariate composed polynomials

AbstractThe Cayley–Dixon formulation for multivariate projection operators (multiples of resultants of multivariate polynomials) has been shown to be efficient (both experimentally and theoretically) for simultaneously eliminating many variables from a polynomial system. In this paper, the behavior of the Cayley–Dixon projection operator and the structure of Dixon matrices are analyzed for composed polynomial systems constructed from a multivariate system in which each variable is substituted by a univariate polynomial in a distinct variable. Under some conditions, it is shown that a Dixon projection operator of the composed system can be expressed as a power of the resultant of the outer polynomial system multiplied by powers of the leading coefficients of the univariate polynomials substituted for variables in the outer system. A new resultant formula is derived for systems where it is known that the Cayley–Dixon construction does not contain any extraneous factor. The complexity of constructing Dixon matrices and roots at toric infinity of composed polynomials is analyzed