Artículo
Abstract:
We study systems of three bivariate polynomials whose Newton polygons are scaled copies of a single polygon. Our main contribution is to construct square resultant matrices, which are submatrices of those introduced by Cattani et al. (1998), and whose determinants are nontrivial multiples 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 our attention to matrices of (almost) Sylvester-type and systems as specified above, then the algorithm yields the smallest possible matrix in general. This is achieved by strongly exploiting the combinatorics of sparse elimination, namely by a new piecewise-linear lifting. The major motivation comes from systems encountered in geometric modeling. Our preliminary Maple implementation, applied to certain examples, illustrates our construction and compares it with alternative matrices. © 2002 Elsevier Science Ltd. All rights reserved.
Registro:
| Documento: | Artículo |
| Título: | Hybrid sparse resultant matrices for bivariate polynomials |
| Autor: | D'Andrea, C.; Emiris, I.Z. |
| Filiación: | Departamento De Matemática, FCEyN, UBA, (1428) Buenos Aires, Argentina INRIA, B.P. 93, Sophia-Antipolis 06902, France |
| Año: | 2002 |
| Volumen: | 33 |
| Número: | 5 |
| Página de inicio: | 587 |
| Página de fin: | 608 |
| DOI: | http://dx.doi.org/10.1006/jsco.2002.0524 |
| Título revista: | Journal of Symbolic Computation |
| Título revista abreviado: | J. Symb. Comput. |
| ISSN: | 07477171 |
| PDF: | https://bibliotecadigital.exactas.uba.ar/download/paper/paper_07477171_v33_n5_p587_DAndrea.pdf |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_07477171_v33_n5_p587_DAndrea |
Referencias:
- Aries, F., Senoussi, R., An implicitization algorithm for rational surfaces with no base points (2001) J. Symb. Comput, 31, pp. 357-365
- Canny, J.F., Emiris, I.Z., A subdivision-based algorithm for the sparse resultant (2000) J. ACM, 47, pp. 417-451. , (May) 3
- Canny, J., Pedersen, P., An algorithm for the Newton resultant (1993), Tech. Rep. 1394, Comp. Science Dept Cornell University; Cattani, E., Cox, D., Dickenstein, A., Residues in toric varieties (1997) Composit. Math, p. 108
- 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., Using Algebraic Geometry, of Graduate Texts in Mathematics (1998), 185. , New York, Springer; D'Andrea, C., Macaulay-style formulas for the sparse resultant (2002) Trans. AMS, , to appear
- D'Andrea, C., Emiris, I.Z., Hybrid resultant matrices of bivariate polynomials (2001) ACM Intern. Symp. on Symbolic and Algebraic Computation, pp. 24-31. , ACM Press
- Emiris, I.Z., On the complexity of sparse elimination (1996) J. Complexity, 12, pp. 134-166
- Gelfand, I., Kapranov, M., Zelevinsky, A., (1994) Discriminants and Resultants, , Boston, MA, Birkhäuser
- Jouanolou, J.-P., Formes d'inertie et résultant: Un formulaire (1997) Adv. Math, 126, pp. 119-250
- Khetan, A., Determinantal formula for the Chow form of a toric surface Proc. ACM Intern. Symp. on Symbolic and Algebraic Computation, , ACM Press, to appear
- Manocha, D., Algebraic and numeric techniques for modeling and robotics (1992), (May), Ph.D. Thesis, Comp. Science Div., Department of Electrical Engineering and Computer Science, University of California, Berkeley; Sturmfels, B., On the Newton polytope of the resultant (1994) J. Algebr. Combin, 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, pp. 301-308. , ACM Press
Citas:
---------- APA ----------
D'Andrea, C. & Emiris, I.Z. (2002) . Hybrid sparse resultant matrices for bivariate polynomials. Journal of Symbolic Computation, 33(5), 587-608.http://dx.doi.org/10.1006/jsco.2002.0524
---------- CHICAGO ----------
D'Andrea, C., Emiris, I.Z. "Hybrid sparse resultant matrices for bivariate polynomials" . Journal of Symbolic Computation 33, no. 5 (2002) : 587-608.http://dx.doi.org/10.1006/jsco.2002.0524
---------- MLA ----------
D'Andrea, C., Emiris, I.Z. "Hybrid sparse resultant matrices for bivariate polynomials" . Journal of Symbolic Computation, vol. 33, no. 5, 2002, pp. 587-608.http://dx.doi.org/10.1006/jsco.2002.0524
---------- VANCOUVER ----------
D'Andrea, C., Emiris, I.Z. Hybrid sparse resultant matrices for bivariate polynomials. J. Symb. Comput. 2002;33(5):587-608.http://dx.doi.org/10.1006/jsco.2002.0524
