Algorithms in Algebraic Geometry

In the last decade, there has been a burgeoning of activity in the design and implementation of algorithms for algebraic geometric compuation. Some of these algorithms were originally designed for abstract algebraic geometry, but now are of interest for use in applications and some of these algorithms were originally designed for applications, but now are of interest for use in abstract algebraic geometry.

The workshop on Algorithms in Algebraic Geometry that was held in the framework of the IMA Annual Program Year in Applications of Algebraic Geometry by the Institute for Mathematics and Its Applications on September 18-22, 2006 at the University of Minnesota is one tangible indication of the interest. One hundred ten participants from eleven countries and twenty states came to listen to the many talks; discuss mathematics; and pursue collaborative work on the many faceted problems and the algorithms, both symbolic and numberic, that illuminate them.

This volume of articles captures some of the spirit of the IMA workshop.

Application of a Numerical Version of Terr Acini’s Lemma for Secants and Joins.- On the sharpness of fewnomial bounds and the number of components of fewnomial hypersurfaces.- Intersections of Schubert varieties and other permutation array schemes.- Efficient Inversion of Rational Maps Over Finite Fields.- Higher-Order Deflation for Polynomial Systems With Isolated Singular Solutions.- Polars of Real Singular Plane Curves.- Semidefinite Representation of the k-Ellipse.- Solving Polynomial Systems Equation by Equation.