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                    Grand Challenges in Computer Algebra
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               The Six-Line Problem and Polynomial Resultants*

     The "Six-Line Problem" arises in the automated analysis of images. A
     three-dimensional object is photographed and from the photographs (or
     other means) a set of geometric invariants are obtained by techniques
     of algebraic geometry. These three-dimensional invariants represent six
     lines abstracted from the object. Suppose that later an object is
     encountered and a photograph taken. The problem is to decide if this is
     the same object as before. From the flat photograph, two-dimensional
     invariants may be obtained.

     The relationship between the three- and two-dimensional invariants may
     be expressed as a set of polynomial equations. The photographed object
     is the right one if the set of equations has a solution. One well known
     method to solve such sets of equations is with resultants. The number
     of variables and the size of the matrix seemed overwhelming until
     recently. This talk will describe how the solution was obtained using
     the Dixon-Kapur-Saxena-Yang resultant.

     Robert H. Lewis
     Department of Mathematics
     Fordham University
     rlewis@murray.fordham.edu

     *Research supported by the Office of Naval Research.