Inversion of parameterized hypersurfaces by means of subresultants

ABSTRACT

We present a subresultant-based algorithm for deciding if the parametrization of a toric hypersurface is invertible or not, and for computing the inverse of the parametrization in the case where it exists. The algorithm takes into account the monomial structure of the input polynomials.

REFERENCES

• 1: Laurent Busé, Residual resultant over the projective plane and the implicitization problem, Proceedings of the 2001 international symposium on Symbolic and algebraic computation, p.48-55, July 2001, London, Ontario, Canada [doi>10.1145/384101.384109]
• 2: Busé, Laurent; Elkadi, Mohamed; Mourrain Bernard. Using projection operators in computer aided geometric design. In Topics in Algebraic Geometry and Geometric Modeling. AMS Press, Contemporary Mathematics 334, 2003.
• 3: John F. Canny , Ioannis Z. Emiris, A subdivision-based algorithm for the sparse resultant, Journal of the ACM (JACM), v.47 n.3, p.417-451, May 2000 [doi>10.1145/337244.337247]
• 4: Cattani, Eduardo; Dickenstein, Alicia; Sturmfels, Bernd. Residues and resultants. J. Math. Sci. Univ. Tokyo 5 (1998), no. 1, 119--148.
• 5: Chardin, Marc. Multivariate subresultants. J. Pure Appl. Algebra 101, no. 2, 129-138 (1995).
• 6: Eng-Wee Chionh , Ronald N. Goldman, Degree, multiplicity, and inversion formulas for rational surfaces using u-resultants, Computer Aided Geometric Design, v.9 n.2, p.93-108, June 1992 [doi>10.1016/0167-8396(92)90009-E]
• 7: Cox, David A. The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4 (1995), no. 1, 17--50.
• 8: Cox, David A. Toric residues. Ark. Mat. 34 (1996), no. 1, 73--96.
• 9: Cox, David A. Equations of parametric curves and surfaces via syzygies. Symbolic computation: solving equations in algebra, geometry, and engineering (South Hadley, MA, 2000), 1--20, Contemp. Math., 286, Amer. Math. Soc., Providence, RI, 2001.
• 10: Cox, David; Little, John; O'Shea, Donal. Using algebraic geometry. Graduate Texts in Mathematics, 185. Springer-Verlag, New York, 1998.
• 11: Carlos D'Andrea , Ioannis Z. Emiris, Hybrid sparse resultant matrices for bivariate polynomials, Journal of Symbolic Computation, v.33 n.5, p.587-608, May 2002 [doi>10.1006/jsco.2002.0524]
• 12: D'Andrea, Carlos; Khetan, Amit. Macaulay style formulas for computing residues. Preprint, 2003 math.AG/030715.
• 13: D'Andrea, Carlos; Khetan, Amit. Implicitization of rational surfaces with toric varieties. Preprint, 2003.
• 14: Gel'fand, I. M.; Kapranov, M. M.; Zelevinsky, A. V. Discriminants, resultants, and multidimensional determinants. Mathematics: Theory & Applications. Birkhauser Boston, Inc., Boston, MA, 1994.
• 15: Handbook of computer aided geometric design. Edited by Gerald Farin, Josef Hoschek and Myung-Soo Kim. North-Holland, Amsterdam, 2002.
• 16: Parametric algebraic curves and applications. Including papers from the IMACS-ACA Session on Parametric Curves and Applications in Computer-aided Geometric Design held at the University of New Mexico, Albuquerque, NM, May 1995. Edited by C. M. Hoffmann, J. R. Sendra and F. Winkler. J. Symbolic Comput. 23 (1997), no. 2--3. Academic Press, Oxford, 1997. pp. 133--333.
• 17: Pérez-Díiaz, Sonia; Schicho, Josef; Sendra, J. Rafael. Properness and inversion of rational parametrizations of surfaces. Appl. Algebra Engrg. Comm. Comput. 13 (2002), no. 1, 29--51.
• 18: Schicho, Josef. Inversion of birational maps with Gröbner bases. Gröbner bases and applications (Linz, 1998), 495--503, London Math. Soc. Lecture Note Ser., 251, Cambridge Univ. Press, Cambridge, 1998.
• 19: J. Rafael Sendra , Franz Winkler, Symbolic parametrization of curves, Journal of Symbolic Computation, v.12 n.6, p.607-631, Dec. 1991 [doi>10.1016/S0747-7171(08)80144-7]
• 20: Bernd Sturmfels, On the Newton Polytope of the Resultant, Journal of Algebraic Combinatorics: An International Journal, v.3 n.2, p.207-236, April 1994 [doi>10.1023/A:1022497624378]
• 21: Szanto, Agnes. Multivariate subresultants using Jouanolou's resultant matrices. Preprint.
• 22: Szanto, Agnes. Solving overdetermined systems by subresultant methods. Preprint.
• 23: Mark van Hoeij, Rational parametrizations of algebraic curves using a canonical divisor, Journal of Symbolic Computation, v.23 n.2-3, p.209-227, Feb./March 1997 [doi>10.1006/jsco.1996.0084]

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.0 GENERAL

General Terms:
Algorithms

Keywords:
birational maps, multivariate subresultants