Topics in Algebraic Geometry and Geometric Modeling

The Gelfond-Khovanskii residue formula computes the sum of the values of any Laurent polynomial over solutions of a system of Laurent polynomial equations whose Newton polytopes have sufficiently general relative position. We discuss two important consequences of this result: an explicit elimination algorithm for such systems and a new formula for the mixed volume. The integer coefficients that appear in the Gelfond-Khovanskii residue formula are geometric invariants that depend only on combinatorics of the polytopes. We explain how to compute them explicitly.

[1]  J. Rafael Sendra,et al.  Parametrization of approximate algebraic surfaces by lines , 2005, Comput. Aided Geom. Des..

[2]  J. Rafael Sendra,et al.  Parametrization of approximate algebraic curves by lines , 2004, Theor. Comput. Sci..

[3]  C. Moura.  Local intersections of plane algebraic curves , 2003 .

[4]  YingLiang Ma,et al.  Point inversion and projection for NURBS curve and surface: Control polygon approach , 2003, Comput. Aided Geom. Des..

[5]  Ivan Soprunov.  Residues and tame symbols on toroidal varieties , 2002, Compositio Mathematica.

[6]  Thomas W. Sederberg,et al.  Approximate Implicitization Using Monoid Curves and Surfaces , 1999, Graph. Model. Image Process..

[7]  G. Ishikawa.  Topological Classification of the Tangent Developables of Space Curves , 1996 .

[8]  Falai Chen,et al.  Implicitization using moving curves and surfaces , 1995, SIGGRAPH.

[9]  Tomoyuki Nishita,et al.  Ray tracing trimmed rational surface patches , 1990, SIGGRAPH.

[10]  D. Mond.  SINGULARITIES OF THE TANGENT DEVELOPABLE SURFACE OF A SPACE CURVE , 1989 .

[11]  Horst Knörrer,et al.  Topologische Typen reeller kubischer Flächen , 1987 .

[12]  T. Urabe.  On quartic surfaces and sextic curves with singularities of type E_8,T_[2,3,7],E_12 , 1984 .

[13]  Kimio Watanabe,et al.  On the Classification of Quartic Surfaces with a Triple Point Part II , 1982 .

[14]  D. Mond.  On the tangent developable of a space curve , 1982, Mathematical Proceedings of the Cambridge Philosophical Society.

[15]  Yumiko Umezu.  On Normal Projective Surfaces with Trivial Dualizing Sheaf , 1981 .

[16]  Ragni Piene,et al.  Cuspidal projections of space curves , 1981 .

[17]  J. P. Cleave,et al.  The form of the tangent-developable at points of zero torsion on space curves , 1980, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  C. Wall,et al.  On the Classification of Cubic Surfaces , 1979 .

[19]  Henry B. Laufer.  On Minimally Elliptic Singularities , 1977 .

[20]  Vladimir I. Arnold,et al.  CRITICAL POINTS OF SMOOTH FUNCTIONS AND THEIR NORMAL FORMS , 1975 .

[21]  Vladimir I. Arnold,et al.  NORMAL FORMS OF FUNCTIONS IN NEIGHBOURHOODS OF DEGENERATE CRITICAL POINTS , 1974 .

[22]  H. G. Telling,et al.  The Rational Quartic Curve in Space of Three and Four Dimensions: Being An Introduction To Rational Curves , 1936 .

[23]  K. Rohn.  Ueber die Flächen vierter Ordnung mit dreifachem Punkte , 1884 .

[24]  Duco van Straten,et al.  A Visual Introduction to Cubic Surfaces Using the Computer Software Spicy , 2003, Algebra, Geometry, and Software Systems.

[25]  R. Goldman,et al.  Topics in algebraic geometry and geometric modeling : Workshop on Algebraic Geometry and Geometric Modeling, July 29-August 2, 2002, Vilnius University, Vilnius, Lithuania , 2003 .

[26]  Askold Khovanskii,et al.  Toric Geometry and Grothendieck Residues , 2002 .

[27]  Michiel Smid,et al.  Closest-Point Problems in Computational Geometry , 2000, Handbook of Computational Geometry.

[28]  R. Piene.  Polar classes of singular varieties , 1978 .

[29]  Beniamino Segre,et al.  The Non-Singular Cubic Surfaces , 1942 .