Artículo

El editor solo permite decargar el artículo en su versión post-print desde el repositorio. Por favor, si usted posee dicha versión, enviela a
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

We unveil in concrete terms the general machinery of the syzygy-based algorithms for the implicitization of rational surfaces in terms of the monomials in the polynomials defining the parametrization, following and expanding our joint article with M. Dohm. These algebraic techniques, based on the theory of approximation complexes due to J. Herzog, A. Simis and W. Vasconcelos, were introduced for the implicitization problem by J.-P. Jouanolou, L. Busé, and M. Chardin. Their work was inspired by the practical method of moving curves, proposed by T. Sederberg and F. Chen, translated into the language of syzygies by D. Cox. Our aim is to express the theoretical results and resulting algorithms into very concrete terms, avoiding the use of the advanced homological commutative algebraic tools which are needed for their proofs. © 2015 Elsevier Ltd.

Registro:

Documento: Artículo
Título:Implicitization of rational hypersurfaces via linear syzygies: A practical overview
Autor:Botbol, N.; Dickenstein, A.
Filiación:Departamento de Matemática, FCEN, Universidad de Buenos Aires, and IMAS-CONICET, Buenos Aires, Argentina
Palabras clave:Implicitization; Matrix representation; Rational surface; Sparse polynomial; Syzygy
Año:2016
Volumen:74
Página de inicio:493
Página de fin:512
DOI: http://dx.doi.org/10.1016/j.jsc.2015.09.001
Título revista:Journal of Symbolic Computation
Título revista abreviado:J. Symb. Comput.
ISSN:07477171
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_07477171_v74_n_p493_Botbol

Referencias:

  • Adkins, W.A., Hoffman, J.W., Wang, H., Equations of parametric surfaces with base points via syzygies (2005) J. Symb. Comput., 39 (1), pp. 73-101
  • Aruliah, D.A., Corless, R.M., Gonzalez-Vega, L., Shakoori, A., Geometric applications of the Bezout matrix in the Lagrange basis (2007) Proceedings of the 2007 International Workshop on Symbolic-Numeric Computation, pp. 55-64. , ACM
  • The implicitization problem for ϕ:Pn[U+21E2](P1)n+1 (2009) J. Algebra, 322 (11), pp. 3878-3895
  • An algorithm for computing implicit equations of bigraded rational surfaces, , arxiv:1007.3690
  • Compactifications of rational maps, and the implicit equations of their images (2011) J. Pure Appl. Algebra, 215 (5), pp. 1053-1068
  • The implicit equation of a multigraded hypersurface (2011) J. Algebra, 348 (1), pp. 381-401
  • Botbol, N., Chardin, M., Castelnuovo-Mumford regularity with respect to multigraded ideals (2015) J. Algebra
  • Botbol, N., Dohm, M., A package for computing implicit equations of parametrizations from toric surfaces, , arxiv:1001.1126
  • Botbol, N., Dickenstein, A., Dohm, M., Matrix representations for toric parametrizations (2009) Comput. Aided Geom. Des., 26 (7), pp. 757-771
  • Botbol, N., Busé, L., Chardin, M., Fitting ideals and multiple-points of surface parameterizations (2014) J. Algebra, 420, pp. 486-508
  • Busé, L., (2006) Elimination theory in codimension one and applications, p. 47. , INRIA research report 5918. Notes of lectures given at the CIMPA-UNESCO-IRAN school in Zanjan, Iran, July 9-22 2005
  • Busé, L., Implicit matrix representations of rational Bézier curves and surfaces (2014) Comput. Aided Des., 46, pp. 14-24. , 2013 {SIAM} Conference on Geometric and Physical Modeling
  • Busé, L., Chardin, M., Implicitizing rational hypersurfaces using approximation complexes (2005) J. Symb. Comput., 40 (4-5), pp. 1150-1168
  • Busé, L., D'Andrea, C., Singular factors of rational plane curves (2012) J. Algebra, 357, pp. 322-346
  • Busé, L., Dohm, M., Implicitization of bihomogeneous parametrizations of algebraic surfaces via linear syzygies (2007) ISSAC 2007, pp. 69-76. , ACM, New York
  • Busé, L., Jouanolou, J.-P., On the closed image of a rational map and the implicitization problem (2003) J. Algebra, 265 (1), pp. 312-357
  • Busé, L., Cox, D.A., D'Andrea, C., Implicitization of surfaces in P3 in the presence of base points (2003) J. Algebra Appl., 2 (2), pp. 189-214
  • Busé, L., Chardin, M., Jouanolou, J.-P., Torsion of the symmetric algebra and implicitization (2009) Proc. Am. Math. Soc., 137 (6), pp. 1855-1865
  • Busé, L., Chardin, M., Simis, A., Elimination and nonlinear equations of Rees algebras (2010) J. Algebra, 324 (6), pp. 1314-1333. , With an appendix in French by Joseph Oesterlé
  • Chardin, M., Implicitization using approximation complexes (2006) Math. Vis., pp. 23-35. , Springer, Berlin, Algebraic Geometry and Geometric Modeling
  • Chen, F., Cox, D.A., Liu, Y., The μ-basis and implicitization of a rational parametric surface (2005) J. Symb. Comput., 39 (6), pp. 689-706
  • Chionh, E.-W., Goldman, R.N., Degree, multiplicity, and inversion formulas for rational surfaces using u-resultants (1992) Comput. Aided Geom. Des., 9 (2), pp. 93-108
  • Commutative Algebra and Its Connections to Geometry (2011) Contemp. Math., 555. , American Mathematical Society, A. Corso, C. Polini (Eds.)
  • Cox, D.A., Equations of parametric curves and surfaces via syzygies (2001) Contemp. Math., 286, pp. 1-20. , Amer. Math. Soc., Providence, RI, Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering
  • Cox, D.A., Curves, surfaces, and syzygies (2003) Contemp. Math., 334, pp. 131-150. , Amer. Math. Soc., Providence, RI, Topics in Algebraic Geometry and Geometric Modeling
  • Cox, D.A., What is a toric variety? (2003) Contemp. Math., 334, pp. 203-223. , Amer. Math. Soc., Providence, RI, Topics in Algebraic Geometry and Geometric Modeling
  • Cox, D.A., Little, J.B., O'Shea, D., Using Algebraic Geometry (1998) Graduate Texts in Mathematics, 185. , Springer-Verlag, New York
  • Cox, D.A., Sederberg, T.W., Chen, F., The moving line ideal basis of planar rational curves (1998) Comput. Aided Geom. Des., 15 (8), pp. 803-827
  • Cox, D.A., Little, J.B., Schenck, H.K., (2011) Toric Varieties, , American Mathematical Society (AMS), Providence, RI
  • Cox, D.A., Kustin, A.R., Polini, C., Ulrich, B., A study of singularities on rational curves via syzygies (2013) Mem. Am. Math. Soc., 222 (1045)
  • D'Andrea, C., Resultants and moving surfaces (2001) J. Symb. Comput., 31 (5), pp. 585-602
  • Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H., Singular 3-1-6 - a computer algebra system for polynomial computations, , http://www.singular.uni-kl.de
  • Diaz-Toca, G.M., Fioravanti, M., Gonzalez-Vega, L., Shakoori, A., Using implicit equations of parametric curves and surfaces without computing them: polynomial algebra by values (2013) Comput. Aided Geom. Des., 30 (1), pp. 116-139
  • Dietz, U., Creation of fair b-spline surface fillets (1998) Creating Fair and Shape Preserving Curves and Surfaces, , BG Teubner, Stuttgart, 2(3), 8
  • Dixon, A.L., The eliminant of three quantics in two independent variables (1908) Proc. Lond. Math. Soc., 7 S2 (1), p. 49
  • Ehrhart, E., Sur un probleme de géométrie diophantienne linéaire ii (1967) J. Reine Angew. Math., 227 (25)
  • Eisenbud, D., Instant elimination, powers of ideals and an Oberwolfach example (2004) Oberwolfach Rep., 1, pp. 1716-1718
  • Fulton, W., Introduction to Toric Varieties (1993) Annals of Mathematics Studies, 131. , Princeton University Press, Princeton, NJ, The William H. Roever Lectures in Geometry
  • Gel'fand, I.M., Kapranov, M.M., Zelevinsky, A.V., (1994) Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications, , Birkhäuser Boston Inc., Boston, MA
  • Grayson, D.R., Stillman, M.E., Macaulay 2, a software system for research in algebraic geometry, , http://www.math.uiuc.edu/Macaulay2/
  • Henrion, D., Sebek, M., Reliable numerical methods for polynomial matrix triangularization (1999) IEEE Trans. Autom. Control, 44 (3), pp. 497-508
  • Herzog, J., Simis, A., Vasconcelos, W.V., Approximation complexes of blowing-up rings (1982) J. Algebra, 74 (2), pp. 466-493
  • Herzog, J., Simis, A., Vasconcelos, W.V., Approximation complexes of blowing-up rings. II (1983) J. Algebra, 82 (1), pp. 53-83
  • Herzog, J., Simis, A., Vasconcelos, W.V., Koszul homology and blowing-up rings (1983) Lecture Notes in Pure and Appl. Math., 84, pp. 79-169. , Dekker, New York, Commutative Algebra
  • Hilbert, D., Ueber die Theorie der Algebraischen Formen (1890) Math. Ann., 36 (4), pp. 473-534
  • Hoffman, W.J., Wang, H., Jia, X., Goldman, R., Minimal generators for the Rees algebra of rational space curves of type (1, 1, d-2) (2010) Eur. J. Pure Appl. Math., 3 (4), pp. 602-632
  • Hoffmann, C., (1989) Geometric Solid Modeling: An Introduction, , Morgan Kaufmann Publishers
  • Jia, X., Goldman, R., μ-bases and singularities of rational planar curves (2009) Comput. Aided Geom. Des., 26 (9), pp. 970-988
  • Jia, X., Wang, H., Goldman, R., Set-theoretic generators of rational space curves (2010) J. Symb. Comput., 45 (4), pp. 414-433
  • Khetan, A., D'Andrea, C., Implicitization of rational surfaces using toric varieties (2006) J. Algebra, 303 (2), pp. 543-565
  • Manocha, D., Canny, J., A new approach for surface intersection (1991) Proceedings of the First ACM Symposium on Solid Modeling Foundations and CAD/CAM Applications, pp. 209-219. , ACM, New York, NY, USA
  • Meyer, F., Zur Theorie der Reducibeln Ganzen Functionen von n Variabeln (1887) Math. Ann., 30 (1), pp. 30-74
  • Salmon, G., (1958) A treatise on the analytic geometry of three dimensions, 1. , (1862). Revised by R.A.P. Rogers. 7th ed. Edited by C.H. Rowe, Chelsea Publishing Company, New York
  • Sederberg, T., Chen, F., (1995) Implicitization using moving curves and surfaces, 303, pp. 301-308
  • Steiner, J., (1832) Systematische Entwicklung der Abhangigkeit geometrischer Gestalten von einander, , Fincke
  • Sturmfels, B., Yu, J.T., Minimal polynomials and sparse resultants (1994) Zero-Dimensional Schemes, pp. 317-324. , de Gruyter, Berlin
  • Thang, L.B., Busé, L., Mourrain, B., Curve/surface intersection problem by means of matrix representations (2009) Proceedings of the 2009 Conference on Symbolic Numeric Computation, pp. 71-78. , ACM
  • Wang, X., Chen, F., Implicitization, parameterization and singularity computation of Steiner surfaces using moving surfaces (2012) J. Symb. Comput., 47 (1), pp. 733-750
  • Zheng, J., Sederberg, T.W., Chionh, E.-W., Cox, D.A., Implicitizing rational surfaces with base points using the method of moving surfaces (2003) Contemp. Math., 334, pp. 151-168. , Amer. Math. Soc., Providence, RI, Topics in Algebraic Geometry and Geometric Modeling

Citas:

---------- APA ----------
Botbol, N. & Dickenstein, A. (2016) . Implicitization of rational hypersurfaces via linear syzygies: A practical overview. Journal of Symbolic Computation, 74, 493-512.
http://dx.doi.org/10.1016/j.jsc.2015.09.001
---------- CHICAGO ----------
Botbol, N., Dickenstein, A. "Implicitization of rational hypersurfaces via linear syzygies: A practical overview" . Journal of Symbolic Computation 74 (2016) : 493-512.
http://dx.doi.org/10.1016/j.jsc.2015.09.001
---------- MLA ----------
Botbol, N., Dickenstein, A. "Implicitization of rational hypersurfaces via linear syzygies: A practical overview" . Journal of Symbolic Computation, vol. 74, 2016, pp. 493-512.
http://dx.doi.org/10.1016/j.jsc.2015.09.001
---------- VANCOUVER ----------
Botbol, N., Dickenstein, A. Implicitization of rational hypersurfaces via linear syzygies: A practical overview. J. Symb. Comput. 2016;74:493-512.
http://dx.doi.org/10.1016/j.jsc.2015.09.001