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D'Andrea, C.; Emiris, I.Z. "Hybrid sparse resultant matrices for bivariate systems" (2001) Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation (ISSAC 2001):24-31
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Abstract:

Our main contribution is an explicit construction of square resultant matrices, which are submatrices of those introduced by Cattani, Dickenstein and Sturmfels [4]. The determinant is a nontrivial multiple of the sparse (or toric) resultant. The matrix is hybrid in that it contains a submatrix of Sylvester type and an additional row expressing the toric Jacobian. If we restrict attention to such matrices, the algorithm yields the smallest possible matrix in general. This is achieved by strongly exploiting the combinatorics of sparse elimination. The algorithm uses a new piecewise-linear lifting, defined for bivariate systems of 3 polynomials with Newton polygons being scaled copies of a single polygon. The major motivation comes from systems encountered in CAD. Our Maple implementation, applied to certain examples, illustrates our construction and compares with alternative matrices.

Registro:

Documento: Conferencia
Título:Hybrid sparse resultant matrices for bivariate systems
Autor:D'Andrea, C.; Emiris, I.Z.
Ciudad:London, Ont.
Filiación:Departamento de Matemática, FCEyN, UBA (1428), Buenos Aires, Argentina
Palabras clave:Convexity; Mixed subdivision; Piecewise-linear lifting; Sparse resultant matrix; Toric Jacobian; Algorithms; Boundary conditions; Combinatorial mathematics; Computer aided design; Computer simulation; Eigenvalues and eigenfunctions; Geometry; Piecewise linear techniques; Polynomials; Vectors; Bivariate systems; Mixed subdivision; Newton polygons; Piecewise linear lifting; Sparse resultant matrix; Toric Jacobian; Matrix algebra
Año:2001
Página de inicio:24
Página de fin:31
Título revista:Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation (ISSAC 2001)
Título revista abreviado:Proc Int Symp Symbol Algebraic Comput ISSAC
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_NIS02119_v_n_p24_DAndrea

Referencias:

  • Canny, J.F., Emiris, I.Z., A subdivision-based algorithm for the sparse resultant (2000) J. ACM, 47 (3), pp. 417-451. , (May)
  • Canny, J., Pedersen, P., An algorithm for the Newton resultant (1993), Tech. Rep. 1394, Comp. Science Dept., Cornell University; Cattani, E., Dickenstein, A., A global view of residues in the torus (1997) J. Pure & Applied Algebra, 117-118, pp. 119-144
  • Cattani, E., Dickenstein, A., Sturmfels, B., Residues and resultants (1998) J. Math. Sci. Univ. Tokyo, 5, pp. 119-148
  • Cox, D., Little, J., O'Shea, D., (1998) Using Algebraic Geometry, , No. 185 in Graduate Texts in Mathematics. Springer-Verlag, New York
  • D'Andrea, C., Dickenstein, A., Explicit formulas for the multivariate resultant (2001) J. Pure Applied Algebra, , To appear
  • Emiris, I.Z., On the complexity of sparse elimination (1996) J. Complexity, 12, pp. 134-166
  • Emiris, I.Z., Mourrain, B., Matrices in elimination theory (1999) J. Symbolic Computation, Special Issue on Elimination, 28, pp. 3-44
  • Galligo, A., Stillman, M., Self-intersection curves of parametrized surfaces (2001), Univ. of Nice, Manuscript; Gelfand, I., Kapranov, M., Zelevinsky, A., (1994) Discriminants and Resultants, , Birkhäuser, Boston
  • Jouanolou, J.-P., Formes d'inertie et résultant: Un formulaire (1997) Adv. in Math., 126, pp. 119-250
  • Kapur, D., Saxena, T., Sparsity considerations in Dixon resultants (1996) Proc. ACM Symp. Theory of Computing, pp. 184-191
  • Manocha, D., (1992) Algebraic and Numeric Techniques for Modeling and Robotics, , PhD thesis, Comp. Science Div., Dept. of Electrical Engineering and Computer Science, Univ. of California, Berkeley, May
  • Sturmfels, B., On the Newton polytope of the resultant (1994) J. of Algebr. Combinatorics, 3, pp. 207-236
  • Zhang, M., Goldman, R., Rectangular corner cutting and Sylvester A-resultants (2000) Proc. ACM Intern. Symp. on Symbolic and Algebraic Computation, , ACM Press

Citas:

---------- APA ----------
D'Andrea, C. & Emiris, I.Z. (2001) . Hybrid sparse resultant matrices for bivariate systems. Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation (ISSAC 2001), 24-31.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_NIS02119_v_n_p24_DAndrea [ ]
---------- CHICAGO ----------
D'Andrea, C., Emiris, I.Z. "Hybrid sparse resultant matrices for bivariate systems" . Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation (ISSAC 2001) (2001) : 24-31.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_NIS02119_v_n_p24_DAndrea [ ]
---------- MLA ----------
D'Andrea, C., Emiris, I.Z. "Hybrid sparse resultant matrices for bivariate systems" . Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation (ISSAC 2001), 2001, pp. 24-31.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_NIS02119_v_n_p24_DAndrea [ ]
---------- VANCOUVER ----------
D'Andrea, C., Emiris, I.Z. Hybrid sparse resultant matrices for bivariate systems. Proc Int Symp Symbol Algebraic Comput ISSAC. 2001:24-31.
Available from: https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_NIS02119_v_n_p24_DAndrea [ ]