Manfred Minimair

Multivariable Resultants

Graduate course, 30 units per 45 minutes

Goals

Resultants provide an important technique for eliminating variables from systems of multivariate polynomials. They have many applications in diverse fields, such as robotics, computer graphics, biochemistry, etc.
This course introduces the notion of multivariable resultant, presents important properties of resultants and algorithms for their construction, and showcases some applications.

Prerequisites

Basic polynomial algebra

Contents

1. Notion of resultant (4 units)
   1. Ideal-based approach
   2. Algebraic geometry approach
2. Macaulay-style construction (11 units)
   1. Univariate case
   2. Projective resultant
   3. Toric resultant
3. Dixon-style construction (7 units)
   1. Univariate case
   2. Maximal minor construction
   3. Resultants over parametrized domains
4. u-Resultants (2 units)
5. Applications (4 units)
6. Final exam (2 units)

Recommended textbook

David A. Cox, John Little, Donal O'Shea
Using Algebraic Geometry
Springer, 2nd edition, 2005

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