Arthur B. Coble Memorial Lectures
33rd Annual Lecture Series

will present

Aspects of Elimination Theory
March 16-18, 2004

• Tuesday, March 16, 4:00 p.m., 314 Altgeld Hal
  I. Resultants, from 1693 to now

• Wednesday, March 17, 4:00 p.m., 245 Altgeld Hall
  II. Exterior algebra methods in algebraic geometry
  Following this lecture, a reception will be held from 5-6 p.m. in the Third Floor Common Room, Altgeld Hall

  The resultant of two polynomials in one variable was born in the need to decide whether two polynomial equations in one variable have a common root. This need actually motivated the discovery of the determinant of a matrix. Later, regarding two polynomials in several variables as polynomials in the last variable whose coefficiets were polynomials in the first variables, the same tools were used to systematically ``eliminate'' the last variable, simplifying the system.

  Concrete and constructive techniques for doing this formed a central theme of 19th century algebra. More recently, the problem was then re-interpreted as one of finding the image of a map of algebraic varieties. In the 20th century van der Waerden, Chow, Grothendieck and others understood and interpreted the 19th century techniques in a new and apparently non-constructive way. Meantime, the possibility of machine computation led to a new interest in the constructive side of the subject, with applications in robotics and elsewhere. Most recently, these two developments have merged again in a very interesting way: the abstract ideas of Grothendieck are the basis for some of the most practical techniques known.

  The goal of my first two lectures will be to survey these developments. The first lecture is meant to be a very accessible survey of the first part of the story, requiring no special background. The second lecture will go further into the modern development.

• Thursday, March 18, 4:00 p.m., 245 Altgeld Hall
  III. Instant elimination, and torsion in symmetric algebras

  If R is a commutative ring and I is an ideal in R, the subring R[It] of the polynomial ring R[t] in one variable over R is called the Rees algebra of I. It is closely related to the construction of the blow-up in algebraic geometry. The geometric formulation of elimination, the problem of finding the image of a map of varieties, can be re-interpreted in terms of this algebra. An unexpected phenomenon, having to do with torsion in this algebra, sometimes turns this hard problem into an easy one. The phenomenon has many applications, but remains mysterious. I will explain these matters by explaining, in detail, a very classical example, the cubic surface in projective 3-space, which is obtained as the image of the projective plane under the map determined by the 4 cubic curves passing through 6 points in the plane. I will also indicate some of the many beautiful and elementary problems from commutative algebra that remain open in this field. This lecture will be essentially independent of lectures I and II.