"From Quadrangular Sets to the Budget Matroids"
Lyle Ramshaw and Jim Saxe
May, 1995. 162 pages.

The complete quadrilateral is a configuration with six points and four
lines in the projective plane.  It is a classical result that, given
the six roots of three quadratic polynomials, we can use the complete
quadrilateral (or its dual, the complete quadrangle) to test
geometrically whether or not those three quadratics are linearly
dependent.  But does there exist some configuration that provides an
analogous geometric test for the linear dependence of four cubic
polynomials?

Yes, there is such a cubic analog of the complete quadrilateral; it
has twelve points, two lines, and thirteen planes in 3-space.  In
fact, the complete quadrilateral and its cubic analog are just two
members of a large family of intriguing configurations that are
defined in this monograph, using the notion of a "budget".  The study
of these "budget configurations" combines old-fashioned geometry with
modern combinatorics -- from null systems to matroids.

Why didn't the classical geometers discover the budget configurations
long ago?  Perhaps because they required their configurations to have
a certain numeric symmetry that a typical budget configuration doesn't
have.

A videotape (134b) accompanies report 134a:

"Introducing the Budget Configurations"
Lyle Ramshaw
Time 45:57 minutes

The videotape animates the cubic analog of the complete
quadrilateral, as well as some other budget configurations that live
in the plane or in 3-space.  By doing so, it tries both to give
non-mathematicians some hint of these discoveries and to lure
mathematicians into reading the monograph.