Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful theories about the solutions to polynomial equations. Enumerative Geometry is concerned with counting the number of solutions when the polynomials come from a geometric situation and Intersection Theory gives methods to accomplish the enumeration.
We use Macaulay 2 to investigate some problems from enumerative geometry. Besides enumerating solutions to the resulting polynomial systems, which include overdetermined, deficient, and improper systems, we address the important question of real solutions to these geometric problems.
The text contains evaluated Macaulay 2 code to illuminate the discussion.
This is intended as a chapter in a book on applications of Macaulay 2 to
problems in mathematics.
While this chapter is largely expository, the results in the last section
concerning lines tangent to quadrics are new.