**Introduction to Numerical Algebraic Geometry**

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

#### Abstract:

In a 1996 paper, Andrew Sommese and Charles Wampler began developing
a new area, ``Numerical Algebraic Geometry'', which would bear the same
relation to ``Algebraic Geometry'' that ``Numerical Linear Algebra'' bears
to ``Linear Algebra''.
To approximate all isolated solutions of polynomial systems,
numerical path following techniques have been proven reliable and
efficient during the past two decades. In the nineties, homotopy
methods were developed to exploit special structures of the polynomial
system, in particular its sparsity. For sparse systems, the roots
are counted by the mixed volume of the Newton polytopes and computed
by means of polyhedral homotopies.

In Numerical Algebraic Geometry we apply and integrate homotopy
continuation methods to describe solution components of polynomial
systems. In particular, our algorithms extend beyond just finding
isolated solutions to also find all positive dimensional solution
sets of polynomial systems and to decompose these into irreducible
components. These methods can be considered as symbolic-numeric,
or perhaps rather as numeric-symbolic, since numerical methods are
applied to find integer results, such as the dimension and degree
of solution components, and via interpolation, to produce symbolic
results in the form of equations describing the irreducible components.

Applications from mechanical engineering motivated the
development of Numerical Algebraic Geometry. The performance of
our software on several test problems illustrates the effectiveness
of the new methods.

In A. Dickenstein and I.Z. Emiris (Eds.),
* Solving Polynomial Equations: Foundations, Algorithms,
and Applications.* Volume 14 of *Algorithms and Computation in
Mathematics*, Springer-Verlag, pages 339-392, 2005.