"Resurrecting Dixon Resultants"
  Deepak Kapur and Tushar Saxena
  kapur@cs.albany.edu

Deepak Kapur and Tushar Saxena
Department of Computer Science
State University of New York
Albany, NY 12222

In the early 1900's, Dixon generalized Cayley's formulation of
Euler-Bezout-Sylvester's resultant for eliminating a single
variable to simultaneously eliminate two variables. Subsequently,
Dixon showed how this technique could be generalized to eliminate
arbitrarily many variables for a subclass of multivariate
polynomials.  For most algebraic and geometry problems, however,
Dixon's method does not give any useful information about common
zeros.  Dixon's work also got overshadowed by Macaulay's
multivariate resultant system which works in general.

In this talk, Dixon's resultants are reviewed. It will be shown
how Dixon's method can be modified to work on a large class of
problems for simultaneously eliminating arbitrarily many
variables.  Examples of problems done easily using this
generalization will be discussed.  Dixon's formulation can solve
most of these examples in a much smaller time than other
techniques.  Dixon's resultants will be compared with Macaulay
resultants as well as sparse resultants based on recent
experimental results.  New results relating the computation of
Dixon's resultants to the Newton polytopes of the input
polynomials will be reviewed, thus establishing that the Dixon
formulation implicitly exploits the sparsity of the input
polynomials.