Hybrid resultant matrix algorithm based on the sylvester-bezout formulation

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

New methods of computing the projective polynomial resultant based on dixon, jouanolou and jacobian matrices

In elimination theory, particularly when using the matrix method to compute multivariate resultant, the ultimate goal is to derive or construct techniques that give a resultant matrix that is of considerable size with simple entries. At the same time, the method should be able to produce no or less superfluous factors. In this thesis, three different techniques for computing the resultant matrix are presented, namely the Jouanolou-Jacobian method, the Dixon-Jouanolou methods for bivariate polynomials, and their generalizations to the multivariate case. The Dixon-Jouanolou method is proposed based on the existing Jouanolou matrix method which is subjected to bivariate systems. To further extend this method to multivariate systems, the entry formula for computing the Dixon resultant matrix is first generalized. This extended application of the loose entry formula leads to the possibility of generalizing the Dixon-Jouanolou method for the bivariate systems of three polynomials to systems of n+1 polynomials with n variables. In order to implement the Dixon-Jouanolou method on systems of polynomials over the affine and projective space, respectively, the concept of pseudohomogenization is introduced. Each space is subjected to its respective conditions; thus, pseudo-homogenization serves as a bridge between them by introducing an artificial variable. From the computing time analysis of the generalized loose entry formula used in the computation of the Dixon matrix entries, it is shown that the method of computing the Dixon matrix using this approach is efficient even without the application of parallel computations. These results show that the cost of computing the Dixon matrix can be reduced based on the number of additions and multiplications involved when applying the loose entry formula. These improvements can be more pronounced when parallel computations are applied. Further analyzing the results of the hybrid Dixon-Jouanolou construction and implementation, it is found that the Dixon-Jouanolou method had performed with less computational cost with cubic running time in comparison with the running time of the standard Dixon method which is quartic. Another independent construction produced in this thesis is the Jouanolou- Jacobian method which is an improvement of the existing Jacobian method since it avoids multi-polynomial divisions. The Jouanolou-Jacobian method is also able to produce a considerably smaller resultant matrix compared to the existing Jacobian method and is therefore less computationally expensive. Lastly all the proposed methods have considered a systematic way of detecting and removing extraneous factors during the computation of the resultant matrix whose determinant gives the polynomial resultant

New methods of computing the projective polynomial resultant based on dixon, jouanolou and jacobian matrices

In elimination theory, particularly when using the matrix method to compute multivariate resultant, the ultimate goal is to derive or construct techniques that give a resultant matrix that is of considerable size with simple entries. At the same time, the method should be able to produce no or less superfluous factors. In this thesis, three different techniques for computing the resultant matrix are presented, namely the Jouanolou-Jacobian method, the Dixon-Jouanolou methods for bivariate polynomials, and their generalizations to the multivariate case. The Dixon-Jouanolou method is proposed based on the existing Jouanolou matrix method which is subjected to bivariate systems. To further extend this method to multivariate systems, the entry formula for computing the Dixon resultant matrix is first generalized. This extended application of the loose entry formula leads to the possibility of generalizing the Dixon-Jouanolou method for the bivariate systems of three polynomials to systems of n+1 polynomials with n variables. In order to implement the Dixon-Jouanolou method on systems of polynomials over the affine and projective space, respectively, the concept of pseudohomogenization is introduced. Each space is subjected to its respective conditions; thus, pseudo-homogenization serves as a bridge between them by introducing an artificial variable. From the computing time analysis of the generalized loose entry formula used in the computation of the Dixon matrix entries, it is shown that the method of computing the Dixon matrix using this approach is efficient even without the application of parallel computations. These results show that the cost of computing the Dixon matrix can be reduced based on the number of additions and multiplications involved when applying the loose entry formula. These improvements can be more pronounced when parallel computations are applied. Further analyzing the results of the hybrid Dixon-Jouanolou construction and implementation, it is found that the Dixon-Jouanolou method had performed with less computational cost with cubic running time in comparison with the running time of the standard Dixon method which is quartic. Another independent construction produced in this thesis is the Jouanolou- Jacobian method which is an improvement of the existing Jacobian method since it avoids multi-polynomial divisions. The Jouanolou-Jacobian method is also able to produce a considerably smaller resultant matrix compared to the existing Jacobian method and is therefore less computationally expensive. Lastly all the proposed methods have considered a systematic way of detecting and removing extraneous factors during the computation of the resultant matrix whose determinant gives the polynomial resultant

Resultants and Discriminants for Bivariate Tensor-product Polynomials

International audienceOptimal resultant formulas have been systematically constructed mostly for unmixed polynomial systems, that is, systems of polynomials which all have the same support. However , such a condition is restrictive, since mixed systems of equations arise frequently in practical problems. We present a square, Koszul-type matrix expressing the resultant of arbitrary (mixed) bivariate tensor-product systems. The formula generalizes the classical Sylvester matrix of two univariate polynomials, since it expresses a map of degree one, that is, the entries of the matrix are simply coefficients of the input polynomials. Interestingly, the matrix expresses a primal-dual multiplication map, that is, the tensor product of a univariate multiplication map with a map expressing derivation in a dual space. Moreover, for tensor-product systems with more than two (affine) variables, we prove an impossibility result: no universal degree-one formulas are possible, unless the system is unmixed. We present applications of the new construction in the computation of discriminants and mixed discriminants as well as in solving systems of bivariate polynomials with tensor-product structure

Matrix formulae for Resultants and Discriminants of Bivariate Tensor-product Polynomials

International audienceThe construction of optimal resultant formulae for polynomial systems is one of the main areas of research in computational algebraic geometry. However, most of the constructions are restricted to formulae for unmixed polynomial systems, that is, systems of polynomials which all have the same support. Such a condition is restrictive, since mixed systems of equations arise frequently in many problems. Nevertheless, resultant formulae for mixed polynomial systems is a very challenging problem. We present a square, Koszul-type, matrix, the determinant of which is the resultant of an arbitrary (mixed) bivariate tensor-product polynomial system. The formula generalizes the classical Sylvester matrix of two univariate polynomials, since it expresses a map of degree one, that is, the elements of the corresponding matrix are up to sign the coefficients of the input polynomials. Interestingly, the matrix expresses a primal-dual multiplication map, that is, the tensor product of a univariate multiplication map with a map expressing derivation in a dual space. In addition we prove an impossibility result which states that for tensor-product systems with more than two (affine) variables there are no universal degree-one formulae, unless the system is unmixed. Last but not least, we present applications of the new construction in the efficient computation of discriminants and mixed discriminants

Loose entry formulas and the reduction of dixon determinant entries

Master'sMASTER OF SCIENC

Multihomogeneous resultant formulae by means of complexes

We provide conditions and algorithmic tools so as to classify and construct the smallest possible determinantal formulae for multihomogeneous resultants arising from Weyman complexes associated to line bundles in products of projective spaces. We also examine the smallest Sylvester-type matrices, generically of full rank, which yield a multiple of the resultant. We characterize the systems that admit a purely B\'ezout-type matrix and show a bijection of such matrices with the permutations of the variable groups. We conclude with examples showing the hybrid matrices that may be encountered, and illustrations of our Maple implementation. Our approach makes heavy use of the combinatorics of multihomogeneous systems, inspired by and generalizing results by Sturmfels-Zelevinsky, and Weyman-Zelevinsky.Comment: 30 pages. To appear: Journal of Symbolic Computatio