﻿ Exact intersection algorithms
Exact intersection algorithms
Modern intersection theory in algebraic geometry has developed from the work of many mathematicians over many decades [Fulton]. It has become a rich and sophisticated theory, yielding a lot of very useful and powerful theorems and techniques. The aim of this work has been is to show how this theory can be used in effective algebraic geometry and practically in Computer Aided Geometric Design (CAGD).

The problems encountered in CAGD are sometimes reminiscent of 19th century problems. At that time, realizing the difficulties one had working in affine  instead of projective space, and over the real numbers instead of the complex numbers one soon shifted the theoretical work towards projective geometry over the complex numbers. In fact, it is still in this situation that the modern intersection theory from algebraic geometry works best.

### Bisection through a Multidimensional Sturm Theorem

A variant of the classical Sturm sequence is presented for computing the number of real solutions of a polynomial system of equations inside a presecribed box. The advantage of this technique is based on the possibility of being used to derive bisection algorithms towards the isolation of the searched real solutions.

### Algortithms for exact interesction

Algorithms using Sturm–Habicht based methods have been implemented and are available at

### References

• [Fulton] W. Fulton, Intersection theory. Springer-Verlag, Berlin-New York 1970.
• [Gerfald] I.M. Gelfand, M. M. Kapranov, A. V. Zelevinski, Discriminants, Resultantsand Multidimensikonal Determinants. Birkh¨auser, Boston 1994.
• [Kharlamov] V. Kharlamov, F. Sottile, Maximally inflected real rational curves, ar-Xiv:math.AG/0206268.

Published May 27, 2005

### Main definitions and theorems

• Intersection multiplicity
• Bezout-type theorems counting intersection points
• Chern and Segre classes

### On the degrees of the resultants

• Classical resultants
• Residual resultants
• Determinantal resultants

### Applications in CAGD

• The degree of a parameterized hypersurface
• Counting intersecting points and curves in P3
• Application to the classication of curves and surfaces
• Rigidity conditions using Chern classes