**An intrinsic homotopy for intersecting
algebraic varieties**

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

#### Abstract:

Recently we developed a diagonal homotopy method to compute a
numerical representation of all positive dimensional components in
the intersection of two irreducible algebraic sets.
In this paper, we rewrite this diagonal homotopy in
intrinsic coordinates, which reduces the number of variables,
typically in half. This has the potential to save a
significant amount of computation, especially in the iterative
solving portion of the homotopy path tracker. Three numerical
experiments all show a speedup of about a factor two.
** 2000 Mathematics Subject Classification.** Primary 65H10;
Secondary 13P05, 14Q99, 68W30.

** Key words and phrases.** Components of solutions, embedding,
generic points, homotopy continuation, irreducible components, numerical
algebraic geometry, polynomial system.

*Journal of Complexity* 21(4): 593-608, 2005.