Arthur D. Chtcherba
ABSTRACT
Matrices constructed from a parameterized multivariate polynomial system are analyzed to ensure that such a matrix contains a condition for the polynomial system to have common solutions irrespective of whether its parameters are specialized or not. Such matrices include resultant matrices constructed using well-known methods for computing resultants over projective, toric and affine varieties. Conditions on these matrices are identified under which the determinant of a maximal minor of such a matrix is a nontrivial multiple of the resultant over a given variety. This condition on matrices allows a generalization of a linear algebra construction, called rank submatrix, for extracting resultants from singular resultant matrices, as proposed by Kapur, Saxena and Yang in ISSAC'94. This construction has been found crucial for computing resultants of non-generic, specialized multivariate polynomial systems that arise in practical applications. The new condition makes the rank submatrix construction based on maximal minor more widely applicable by not requiring that the singular resultant matrix have a column independent of the remaining columns. Unlike perturbation methods, which require introducing a new variable, rank submatrix construction is faster and effective. Properties and conditions on symbolic matrices constructed from a polynomial system are discussed so that the resultant can be computed as a factor of the determinant of a maximal non-singular submatrix.
- F. Arries and R. Senoussi. An Implicitization Algorithm for Rational Surfaces with no Base Points. Journal of Symbolic Computation, 31:357--365, 2001. Google Scholar
Digital Library
- L. Busè, M. Elkadi, and B. Mourrain. Generalized resultants over unirational algebraic varieties. J. Symbolic Computation, 29:515--526, 2000. Google ScholarDigital Library
- L. Busè, M. Elkadi, and B. Mourrain. Resultant over the residual of a complete intersection. Journal of Pure and Applied Algebra, 164(1-2):35--57, 2001.Google ScholarCross Ref
- L. Busè, M. Elkadi, and B. Mourrain. Using projection operators in computer aided geometric design. In Topics in Algebraic Geometry and Geometric Modeling, volume 334 of Contemporary Mathematics, pages 321--342. American Mathematical Society, 2003.Google Scholar
- J. F. Canny and I. Z. Emiris. A subdivision-based algorithm for the sparse resultant. Journal of the ACM, 47(3):417--451, 2000. Google ScholarDigital Library
- M. Chardin. Multivariate subresultants. J. Pure and Appl. Alg., 101:129--138, 1995.Google ScholarCross Ref
- A. D. Chtcherba. A new Sylvester-type Resultant Method based on the Dixon-Bézout Formulation. PhD dissertation, University of New Mexico, Department of Computer Science, Aug 2003. Google ScholarDigital Library
- A. D. Chtcherba and D. Kapur. A Combinatorial Perspective on Constructing Dialytic Resultant Matrices. Proc. of 10th Rhine Workshop (RCWA'06), pages 67--81, 2006.Google Scholar
- D. Cox, J. Little, and D. O'Shea. Using Algebraic Geometry. Springer-Verlag, New York, first edition, 1998.Google Scholar
- C. D'Andrea and I. Z. Emiris. Hybrid sparse resultant matrices for bivariate polynomials. J. Symb. Comput., 33(5):587--608, 2002. Google ScholarDigital Library
- A. Dixon. The eliminant of three quantics in two independent variables. Proc. London Mathematical Society, 6:468--478, 1908.Google ScholarCross Ref
- I. Emiris. Sparse Elimination and Applications in Kinematics. PhD thesis, Department of Computer Science, Univeristy of Calif., Berkeley, 1994. Google ScholarDigital Library
- I. Emiris and J. Canny. Efficient incremental algorithms for the sparse resultant and the mixed volume. J. Symbolic Computation, 20(2):117--149, August 1995. Google ScholarDigital Library
- I. Emiris and B. Mourrain. Matrices in elimination theory. Journal of Symbolic Computation, 28(1-2):3--43, 1999. Google ScholarDigital Library
- I. Gelfand, M. Kapranov, and A. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Birkhauser, Boston, first edition, 1994.Google Scholar
- H. Hong, R. Liska, and S. Steinberg. Testing stability by quantifier elimination. J. Symb. Comput., 24(2):161--187, 1997. Google ScholarDigital Library
- J. Jouanolou. Le formalisme du résultant. Adv. in Math., 90:117--263, 1991.Google ScholarCross Ref
- D. Kapur, T. Saxena, and L. Yang. Algebraic and geometric reasoning using the Dixon resultants. In ISSAC, pages 99--107, Oxford, England, Jul 1994. Google ScholarDigital Library
- A. Khetan. The resultant of an unmixed bivariate system. Journal of Symbolic Computation, 36(3-4):425--442, 2003. Google ScholarDigital Library
- D. Manocha and J. F. Canny. Implicit representation of rational parametric surfaces. Journal of Symbolic Computation, 13(5):485--510, 1992. Google ScholarDigital Library
- P.Pedersen and B. Sturmfels. Product formulas for resultants and chow forms. Math. Zeitschrift, 214:377--396, 1993.Google ScholarCross Ref
- T. Saxena. Efficient variable elimination using resultants. PhD thesis, Department of Computer Science, State Univeristy of New York, Albany, NY, 1997. Google ScholarDigital Library
- I. Shafarevich. Basic Algebraic Geometry. Spring-Verlag, New-York, second edition, 1994. Google ScholarDigital Library
- J. Zheng, T. W. Sederberg, E.-W. Chionh, and D. A. Cox. Implicitizing rational surfaces with base points using the method of moving surfaces. In Topics in Algebraic Geometry and Geometric Modeling, volume 334 of Contemporary Mathematics, pages 151--168. American Mathematical Society, 2003.Google Scholar
Index Terms
- Conditions for determinantal formula for resultant of a polynomial system
Recommendations
Finding all real zeros of polynomial systems using multi-resultant
We present a new practicable method for approximating all real zeros of polynomial systems using the resultants method. It is based on the theory of multi-resultants. We build a sparse linear system. Then, we solve it by the quasi-minimal residual ...
Multihomogeneous resultant formulae for systems with scaled support
ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computationConstructive methods for matrices of multihomogeneous resultants for unmixed systems have been studied in [7, 13, 15]. We generalize these constructions to mixed systems, whose Newton polytopes are scaled copies of one polytope, thus taking a step ...
Heuristics to accelerate the Dixon resultant
The Dixon resultant method solves a system of polynomial equations by computing its resultant. It constructs a square matrix whose determinant (det) is a multiple of the resultant (res). The naive way to proceed is to compute det, factor it, and ...
Comments