**Solving Polynomial Systems Equation by Equation**

*Andrew J. Sommese*,
*Jan Verschelde*, and
*Charles W. Wampler*

#### Abstract:

By a numerical continuation method called a diagonal
homotopy, one can compute the intersection of two irreducible
positive dimensional solution sets of polynomial systems. This
paper proposes to use this diagonal homotopy as the key step in a
procedure to intersect general solution sets that are not
necessarily irreducible or even equidimensional. Of particular
interest is the special case where one of the sets is defined by a
single polynomial equation. This leads to an algorithm for finding a
numerical representation of the solution set of a system of
polynomial equations introducing the equations one by one.
Preliminary computational experiments show this approach can exploit
the special structure of a polynomial system, which improves the
performance of the path following algorithms.
** 2000 Mathematics Subject Classification.** Primary 65H10;
Secondary 13P05, 14Q99, 68W30.

** Key words and phrases.** Algebraic set, components of solutions,
diagonal homotopy, embedding, equation-by-equation solver, generic point,
homotopy continuation, irreducible component, numerical irreducible
decomposition, numerical algebraic geometry, path following, polynomial
system, witness point, witness set.