Hybrid resultant matrix algorithm based on the sylvester-bezout formulation

Abstract

The resultant of a system of polynomial equations is a factor of the determinant of the resultant matrix. The matrix is said to be optimal when its determinant equals exactly the resultant. Other factors besides the determinant are known as extraneous factors and it has been the major interest among researches to seek for a determinantal resultant formula that gives optimal resultant matrix whose determinant exactly equals the resultant. If such determinantal formula does not exist, a formulation that reduces the existence of these extraneous factors is sought. This thesis focuses on the construction and implementations of determinantal formulas that gives exact resultant for certain classes of multihomogeneous multivariate polynomial equations. For the class of multigraded polynomial systems, a Sylvester type formula giving exact resultant can be derived out of certain degree vectors. The first part of this thesis implements the Sylvester type formula for determining the entries and dimension of the Sylvester type matrix of multigraded systems by applying the properties of certain linear maps and permutations of groups of variables. Even though the Sylvester type formula gives exact resultants for multigraded systems, this approach does not take advantage of the sparseness conditions when considering sparse polynomials. Sparse systems can be utilized by considering the underlying properties of its Newton polytopes, the convex hull of the support of the system. Preliminary observations on the properties of mixed volumes of the polytopes in comparison to the degree of the resultant of polynomial systems derived from Sylvester type matrices are used in the determination of whether the resultant matrix is optimal. This research proceeds to construct and implement a new hybrid resultant matrix algorithm based on the Sylvester-B´ezout formulation. The basis of this construction applies some related concepts and tools from algebraic geometry such as divisors, fans and cones, homogeneous coordinate rings and the projective space. The major tasks in the construction are determining the degree vector of the homogeneous variables known as homogeneous coordinates and solving a set of linear inequalities. In this work, the method of solving these equations involves a systematic procedure or combinatorial approach on the set of exponent vectors of the monomials. Two new rules are added as a termination criterion for obtaining the unique solutions for the B´ezout matrix. The implementation of the new algorithm on certain class of unmixed multigraded systems of bivariate polynomial equations with some coefficients being zero suggests conditions that can produce exact resultant. From the results, some theorems on these conditions and properties are proven. An application of the hybrid resultant matrix to solving the multivariate polynomial equations in three variables is discussed. Upon completion of this research two new computer algebra packages have been developed, namely the Sylvester matrix package for multivariate polynomial equations and the hybrid Sylvester- B´ezout matrix package for computing the resultant of bivariate polynomial equations.